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+{"text":"\\section{INTRODUCTION}\n\n\nMillisecond pulsars (MSPs) are neutron stars (NSs) with short spin periods ($P\\leq 30$ ms) and weak surface magnetic fields \\citep[$B \\sim 10^{8}-10^{9}$ G;][]{L08}. They are believed to be old NSs that have been reactivated by accretion from their companion stars during the low-mass X-ray binary (LMXB) phase \\citep{R82,A82}. The mass transferred from the companion star not only causes the decay of the NS's magnetic field, but also accelerates the NS's spin period to milliseconds \\citep[see, e.g.,][for reviews]{B91,T06}. This recycling scenario has been strongly supported by the discovery of several X-ray pulsars with millisecond periods in LMXBs \\citep[see][for a summary]{P2012}, and the discovery of the transition between a rotation-powered MSP state and a LMXB state in PSR J1023+0038 \\citep{A09}, IGR J18245-2452 \\citep{P2013}, and PSR J1227-4853 \\citep{R15}.\n\n\nWhen the mass transfer ceases, some of the LMXBs evolve to be binary systems consisting of a MSP and a He white dwarf (WD). This population is called low-mass binary pulsars (LMBPs) \\citep{T06}. Compared with LMBPs, there is another population called intermediate-mass binary pulsars (IMBPs), which contain a pulsar with a spin period of tens of milliseconds and a CO or ONeMg WD with mass $\\geq 0.4M_{\\odot}$ \\citep{C96,C01,C11}.\nMost IMBPs are thought to have evolved from intermediate-mass X-ray binaries (IMXBs). Because of the relatively high mass ratio ($q=M_{2}\/M_{1}>1.5$, where $M_{1}$ and $M_{2}$ are the NS mass and the donor star mass, respectively), it had been suggested that the mass transfer could be unstable, and the NS would spiral into the envelope of the donor in less than a millennium \\citep{P76,W84,I93}. More recent studies showed that X-ray binary systems with an intermediate-mass donor star may avoid entering the spiral-in phase, undergoing rapid mass transfer on a (sub)thermal timescale, and eventually evolve into IMBPs \\citep[e.g.,][]{T00,P00,K2000,S12}. Since the birth rate of IMXBs is significantly larger than that of original LMXBs, and since IMXBs likely evolve to resemble observed LMXBs when the mass ratio becomes less than unity, it is generally believed that the majority of LMXBs may have started their lives with an intermediate-mass donor star \\citep[e.g.,][]{P03}.\n\nPSR J1640+2224 is a MSP with a spin period $P=3.16$ ms. It is located in a wide, nearly circular binary system (orbital period $P_{\\rm orb}=175$ days and orbital eccentricity $e=7.9725\\times 10^{-4}$) with a WD companion \\citep{L96}. These characteristics seem to indicate that this system has evolved from a wide LMXB \\citep{T11b}. According to the theoretical correlation between the orbital period and the WD mass \\citep{R95,T99,L11,J14}, the WD companion's mass would be $\\sim (0.35-0.39)\\,M_{\\odot}$ depending on the metallicity. \\citet{V18} recently presented the first astrometric parallax measurement of PSR J1640+2224, based on the observations taken with the Very Long Baseline Array. Using the new distance in the analysis of the Hubble Space Telescope observation, they found that the WD mass is $0.71^{+0.21}_{-0.20}\\,M_{\\odot}$ and $0.66^{+0.21}_{-0.19}\\,M_{\\odot}$ for DA and DB WDs, respectively. The results indicate that the WD mass is larger than $0.4M_{\\odot}$ with $>90\\%$-confidence, so it is most likely a CO WD. However, previous works predicted that most IMBPs formed from IMXBs have a spin period of tens of milliseconds and an orbital period $P_{\\rm orb}<40$ days \\citep{T00,P00,P02,P03,S12}. These characteristics are in conflict with the observational parameters of PSR J1640+2224. Therefore, the formation of this system had been a puzzle.\n\nIn this paper, we try to explore the formation route of PSR J1640+2224, paying particular attention to the influence of the NS mass and the metallicity of the donor star. The reminder of this  paper is organized as follows. We describe the stellar evolution code and the binary model in Section 2. The calculated binary evolution results are demonstrated in Section 3. We present our discussion and summarize in Section 4.\n\n\n\\section{BINARY EVOLUTIONARY CALCULATIONS}\n\n\\subsection{The stellar evolution code}\nAll the calculations were carried out by using the stellar evolution code Modules for Experiments in Stellar Astrophysics (MESA; version number 11554; Paxton et al. 2011, 2013, 2015, 2018, 2019). We have calculated the evolutions of a large number of I\/LMXBs, adopting both Population \\uppercase\\expandafter{\\romannumeral1} ($X=0.7$ and $Z=0.02$) and \\uppercase\\expandafter{\\romannumeral2} ($X=0.75$ and $Z=0.001$) chemical compositions for the donor stars. For the treatment of convection in the donor stars, we employ both exponential diffusive overshooting with the parameter $f_{\\rm ov}=0.01-0.016$ \\citep{H00} and a mixing-length parameter of $\\alpha=2.0$ (see Section 4). In our calculations, we have considered a number of binary interactions to follow the details of mass transfer processes, including gravitational radiation (GR), magnetic braking (MB) and mass loss, which lead to orbital angular momentum loss.\n\n\n\\subsection{The input physics}\nWe assume that the binary initially consists of a NS of mass $M_{1}$ and a zero-age main-sequence donor of mass $M_{2}$. The Roche-lobe (RL) radius $R_{\\rm L,2}$ of the donor is evaluated with the formula proposed by \\citet{E83},\n\\begin{equation}\n\\frac{R_{\\rm L,2}}{a}=\\frac{0.49 q^{-2 \/ 3}}{0.6 q^{-2 \/ 3}+\\ln \\left(1+q^{-1 \/ 3}\\right)},\n\\end{equation}\nwhere $a$ is the orbital separation of the binary and $q=M_{2}\/M_{1}$ is the mass ratio. We adopt the \\citet{R88} scheme to calculate the mass transfer rate via Roche-lobe overflow (RLOF),\n\\begin{equation}\n-\\dot{M}_{2}=\\dot{M}_{2,0} \\exp \\left(-\\frac{R_{2}-R_{\\mathrm{L}, 2}}{H}\\right),\n\\end{equation}\nwhere $H$ is the scale-height of the atmosphere evaluated at the surface of the donor, $R_{2}$ is the radius of the donor, and\n\\begin{equation}\n\\dot{M}_{2,0}=\\frac{1}{e^{1 \/ 2}} \\rho c_{\\mathrm{th}} Q,\n\\end{equation}\nwhere $\\rho$ and $c_{\\mathrm{th}}$ are the mass density and the sound speed on the surface of the star respectively, and $Q$ is the cross section of the mass flow via the $L_{1}$ point. The very small orbital eccentricity indicates that tides keep the binary orbit circular \\citep{K88}, so the orbital angular momentum is\n\\begin{equation}\nJ_{\\mathrm{orb}}=\\frac{M_{1} M_{2}}{M_{1}+M_{2}} \\Omega a^{2},\n\\end{equation}\nwith the orbital angular velocity $\\Omega=\\sqrt{GM\/a^{3}}$ (where $M=M_1+M_2$ is the total mass). Taking the logarithmic derivative of Eq.~(4) with respect to time gives the rate of change in the orbital separation\n\\begin{equation}\n\\frac{\\dot{a}}{a}=2 \\frac{\\dot{J}_{\\mathrm{orb}}}{J_{\\mathrm{orb}}}-2 \\frac{\\dot{M}_{1}}{M_{1}}-2 \\frac{\\dot{M}_{2}}{M_{2}}+\\frac{\\dot{M}}{M}.\n\\end{equation}\nHere the total rate of change in the orbital angular momentum is determined by\n\\begin{equation}\n\\frac{\\dot{J}_{\\mathrm{orb}}}{J_{\\mathrm{orb}}}=\\frac{\\dot{J}_{\\mathrm{gr}}}{J_{\\mathrm{orb}}}+\\frac{\\dot{J}_{\\mathrm{mb}}}{J_{\\mathrm{orb}}}+\\frac{\\dot{J}_{\\mathrm{ml}}}{J_{\\mathrm{orb}}}.\n\\end{equation}\nThe three terms on the right-hand-side of Eq.~(6) represent angular momentum losses caused by GR, MB, and mass loss, respectively. The GR-induced rate $\\dot{J}_{\\rm gr}$ is calculated with the standard formula \\citep{L59,F71}\n\\begin{equation}\n\\frac{\\dot{J}_{\\mathrm{gr}}}{J_{\\mathrm{orb}}}=-\\frac{32 G^{3}}{5 c^{5}} \\frac{M_{1} M_{2} M}{a^{4}},\n\\end{equation}\nwhere $G$ and $c$ are the gravitational constant and the speed of light, respectively. The prescription of \\citet{V81} is adopted to calculate the angular momentum loss due to MB,\n\\begin{equation}\n\\frac{\\dot{J}_{\\mathrm{mb}}}{J_{\\mathrm{orb}}}=-3.8 \\times 10^{-30} \\frac{G R_{2}^{4} M^{2}}{a^{5} M_{1}} \\mathrm{s}^{-1}.\n\\end{equation}\nFor IMXBs and wide LMXBs, the mass transfer rate $|\\dot{M_{2}}|$ may be higher than the Eddington-limit accretion rate $\\dot{M}_{\\mathrm{Edd}}$ of the NS. Thus, the accretion rate of the NS is evaluated using the following formula:\n\\begin{equation}\n\\dot{M}_{\\mathrm{1}}=\\min \\left(\\left|\\dot{M}_{2}\\right|, \\dot{M}_{\\mathrm{Edd}}\\right),\n\\end{equation}\nand the mass loss rate from the binary system is:\n\\begin{equation}\n\\dot{M}=\\dot{M}_{1}-\\left|\\dot{M}_{2}\\right|.\n\\end{equation}\nIn the case of super-Eddington accretion, we adopt the isotropic reemission model, assuming that the extra material leaves the binary in the form of isotropic wind from the NS. Therefore, the angular momentum loss rate due to mass loss can be derived to be\n\\begin{equation}\n\\dot{J}_{\\mathrm{ml}}=-\\left(\\left|\\dot{M}_{2}\\right|-\\dot{M}_{\\mathrm{1}}\\right) a_{\\mathrm{1}}^{2} \\Omega,\n\\end{equation}\nwhere $a_{\\mathrm{1}}$ is the distance between the NS and the center of mass of the binary.\n\n\n\n\\section{RESULTS OF EVOLUTION CALCULATIONS}\n\nWe have performed calculations of the binary evolution for thousands of I\/LMXB systems. We choose the initial NS mass $1.4\\,M_{\\odot}\\leq M_{1}\\leq 2.2\\,M_{\\odot}$ and the initial donor mass $1.0\\,M_{\\odot}\\leq M_{2}\\leq 4.0\\,M_{\\odot}$, and set the initial orbital period 1.0 day $\\leq P_{\\rm orb}\\leq 60$ days. We use exponential diffusive overshooting for the donor stars.  Note that there exists a bifurcation period for the initial orbital period $P_{\\rm bf}\\sim 1$ day \\citep{Py88,Py89}. If the initial orbital period is below $P_{\\rm bf}$, the binary systems will evolve with shrinking orbits, possibly forming ultra-compact binaries. This kind of evolution cannot reproduce PSR J1640+2224, and will not be considered here. In addition, we exclude the evolutions with the final orbital periods exceeding 500 days.\n\n\n\\begin{figure}\n\n\\centerline{\\includegraphics[scale=0.6]{f1.eps}}\n\\caption{Example evolution of a typical IMXB consisting of a NS and a donor star  with initial masses $M_{1}=1.4M_{\\odot}$ and $M_{2}=3.5M_{\\odot}$ respectively, and orbital period $P_{\\rm orb}=6$ days. The blue and red solid curves denote the evolutionary tracks for the mass transfer rate and the NS mass, respectively, and the blue dashed curve the Eddington-limit accretion rate.\n   \\label{figure1}}\n\n\\end{figure}\n\n\\begin{figure}\n\n\\centerline{\\includegraphics[scale=0.6]{f2.eps}}\n\\caption{Same as Fig.~1. The green and blue curves denote the evolutionary tracks for the orbital period and the donor mass, respectively.\n   \\label{figure2}}\n\n\\end{figure}\n\n\\subsection{A typical example for the IMXB evolution}\nWe present the evolutionary sequences for a typical IMXB system that consists of a $1.4M_{\\odot}$ NS and a $3.5M_{\\odot}$ donor star with an initial orbital period $P_{\\rm orb}=6$ days in Figs.~1 and 2. The donor star starts to fill its RL and commence mass transfer at the age of 240.6 Myr. The mass transfer process lasts about 1.9 Myr, at a rate higher than the Eddington accretion rate (the maximum mass transfer rate $>$ $10^{-5}M_{\\odot}$\\, yr$^{-1}$). In this case, most of the transferred material from the donor is lost from the binary system. Thus, after the mass transfer, the donor evolves into a CO WD of mass $0.543M_{\\odot}$, while the NS has accreted only $0.034M_{\\odot}$ mass. The orbital period becomes smaller in the former stage of the mass transfer process due to the relatively high mass ratio ($>$1); when the mass ratio is reversed, the orbit begins to expand. The final orbital period is $P_{\\rm orb}=17.8$ days.\n\n\\begin{figure}\n\n\\centerline{\\includegraphics[scale=0.6]{f3.eps}}\n\\caption{The final orbital period as a function of the WD mass. The yellow, green, red and blue curves are for the initial NS mass of $1.4M_{\\odot}$,  $1.8M_{\\odot}$, $2.0M_{\\odot}$ and $2.2M_{\\odot}$, respectively. Beside each curve we label the initial donor mass. The gray, solid horizontal line represents the distribution of the orbital period and the WD mass for PSR J1640+2224. The triangles are used to distinguish different curves that overlap. In the top and bottom panels we adopt different chemical compositions and overshooting parameters (top: Population \\uppercase\\expandafter{\\romannumeral1} and $f_{\\rm ov}=0.016$; bottom: Population \\uppercase\\expandafter{\\romannumeral2} and $f_{\\rm ov}=0.01$).\n   \\label{figure3}}\n\n\\end{figure}\n\n\\subsection{The $P_{\\rm orb}^{\\rm final}-M_{\\rm WD}$ diagram}\nFig.~3 shows the calculated final orbital period as a function of the WD mass for different initial masses of the donor star and the NS. The yellow, green, red, and blue curves are for NSs with the initial mass of $1.4M_{\\odot}$, $1.8M_{\\odot}$, $2.0M_{\\odot}$, and $2.2M_{\\odot}$, respectively. Beside each curve we label the initial mass of the donor star. The triangles are used to denote the final systems with longest orbital periods for a give donor star, and also to distinguish different curves that  overlap. The gray horizontal line represents the orbital period $-$ WD mass distribution for PSR J1640+2224.\n\n\nIn the top panel we show the results with Population I chemical compositions. We first examine the orbital period - WD mass relation for a $1.4 M_{\\odot}$ NS. In this case, it is clear to see that, for a given WD mass $> 0.4\\,M_{\\odot}$, the predicted final orbital period is significantly shorter than 175 days. For more massive donor star, the resultant WD is more massive, while the final orbital period is shorter. So they are obviously unable to account for the properties of PSR J1640+2224. If we increase the mass of the NS, the final orbital period becomes longer. When $M_{1}= 2.0\\,M_{\\odot}$, it is possible to simultaneously account for the WD of mass $\\sim 0.4 \\,M_{\\odot}$ and the orbital period $\\sim$ 175 days. Moreover, if the WD is more massive than $0.6\\,M_{\\odot}$, then the initial NS mass should be enhanced to be $2.2\\,M_{\\odot}$.\n\nThe bottom panel shows the results with Population II chemical compositions. It is known that lower metallicities lead to smaller stellar radius and shorter nuclear evolutionary timescale. So stars with the same mass but lower metallicities form WDs in a narrower orbit \\citep{T99,J14}. We can see that the evolution of LMXBs containing a $1.4M_{\\odot}$ NS and a $1.0M_{\\odot}$ donor star may form PSR J1640+2224-like binaries with a $0.4M_{\\odot}$ WD companion. However, if the WD indeed has a mass higher than $0.6M_{\\odot}$, then we still require the NS to be initially more massive than $2.0M_{\\odot}$.\n\nWe summarize our calculated results with $M_{\\rm WD} \\geq 0.4 M_{\\odot}$ and the final orbital period 170 $\\leq P_{\\rm orb} \\leq$ 180 days in Table 1. It is noted that PSR J1640+2224 is most likely to have descended from an IMXB via Case B RLOF. Tables 2-5 present more general results by taking into account different initial NS mass.\n\n\\begin{table}\n\\footnotesize\n\\begin{center}\n\\caption{Parameters for IMXB evolution with the final orbit period 170 days $\\leq P_{\\rm orb}\\leq 180$ days and the WD mass $M_{\\rm WD}\\geq0.4\\,M_{\\odot}$}\n\\begin{tabular}{ccccccccccc}\n  \\toprule[2pt]\n  $M^{\\rm ini}_{1}$ & $M^{\\rm ini}_{2}$ & $P^{\\rm ini}_{\\rm orb}$ & $t_{\\rm RLO}$  & $M^{\\rm fin}_{1}$ & $M^{\\rm fin}_{2}$ & $P^{\\rm fin}_{\\rm orb}$ &  & $\\bigtriangleup M_{1}$ & $\\bigtriangleup t_{\\dot{M}}$ & $\\dot{M}_{\\rm max}$   \\\\\n  ($M_{\\odot}$) & ($M_{\\odot}$) &    (days)       & (Myr)        & ($M_{\\odot}$) & ($M_{\\odot}$) &     (days)      &   Type   &    ($M_{\\odot}$)       &           (Myr)              & ($M_{\\odot}$ yr$^{-1}$)   \\\\\n  (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11)    \\\\\n \n \n \n\n  \\hline\n    \\multicolumn{11}{c}{Population \\uppercase\\expandafter{\\romannumeral1} ($X=0.7$, $Z=0.02$)} \\\\\n  \\hline\n  $2.0$  & 2.5 & 5.35 & 597.900  & 2.109 & 0.414 & 170.0 & HeCO & 0.109 & 5.845 & $3.16\\times 10^{-4}$    \\\\\n  $2.2$  & 2.5 & 7.0 & 598.596  & 2.304 & 0.488 & 175.0 & CO & 0.104 & 6.580 & $5.01\\times 10^{-6}$    \\\\\n  $2.2$  & 3.0 & 7.0 & 363.239 & 2.255 & 0.506 & 176.5 & CO & 0.055 & 3.363 & $7.94\\times 10^{-6}$    \\\\\n  $2.2$  & 3.5 & 10.8 & 241.819  & 2.226 & 0.571 & 171.5 & CO & 0.026 & 1.558 & $2.51\\times 10^{-5}$    \\\\\n  $2.25$  & 4.0 & 15.2 & 170.965  & 2.266 & 0.634 & 170.8 & CO & 0.016 & 0.857 & $6.31\\times 10^{-5}$    \\\\\n  \\hline\n  \\multicolumn{11}{c}{Population \\uppercase\\expandafter{\\romannumeral2} ($X=0.75$, $Z=0.001$)} \\\\\n  \\hline\n  $1.4$  & 1.0 & 29.0 & 5965.290  & 1.732 & 0.400 & 177.9 & He & 0.332 & 19.440 & $7.00\\times 10^{-8}$    \\\\\n  $1.4$  & 1.5 & 17.0 & 1557.870  & 1.837 & 0.400 & 173.9 & He & 0.437 & 27.950 & $3.16\\times 10^{-7}$    \\\\\n  $1.8$  & 1.5 & 12.5 & 1547.698  & 2.427 & 0.400 & 175.4 & He & 0.627 & 38.912 & $7.59\\times 10^{-8}$    \\\\\n  $1.8$  & 2.0 & 12.0 & 642.987  & 2.046 & 0.479 & 171.4 & CO & 0.246 & 13.620 & $1.58\\times 10^{-6}$    \\\\\n  $2.0$  & 1.5 & 11.5 & 1544.370  & 2.694 & 0.400 & 178.3 & He & 0.694 & 42.240 & $5.89\\times 10^{-8}$    \\\\\n  $2.0$  & 2.0 & 10.5 & 650.153  & 2.293 & 0.484 & 174.1 & CO & 0.293 & 16.379 & $5.01\\times 10^{-7}$    \\\\\n  $2.0$  & 2.5 & 10.5 & 370.271  & 2.110 & 0.525 & 174.3 & CO & 0.110 & 6.503 & $7.94\\times 10^{-6}$    \\\\\n \n  $2.2$  & 1.5 & 10.5 & 1540.127  & 2.965 & 0.400 & 176.9 & He & 0.756 & 46.617 & $4.37\\times 10^{-8}$    \\\\\n  $2.2$  & 2.0 & 9.0 & 648.640  & 2.524 & 0.484 & 170.7 & CO & 0.324 & 13.085 & $3.16\\times 10^{-7}$    \\\\\n  $2.2$  & 2.5 & 8.5 & 369.724  & 2.32 & 0.525 & 173.2 & CO & 0.120 & 7.045 & $6.31\\times 10^{-6}$    \\\\\n  $2.2$  & 3.0 & 12.3 & 236.243  & 2.262 & 0.616 & 178.8 & CO & 0.062 & 4.155 & $2.85\\times 10^{-5}$    \\\\\n  $2.2$  & 3.5 & 19.0 & 169.025  & 2.246 & 0.723 & 177.3 & CO & 0.046 & 3.098 & $7.94\\times 10^{-5}$    \\\\\n  $2.2$  & 4.0 & 25.5 & 125.904  & 2.231 & 0.794 & 170.0 & CO & 0.031 & 1.760 & $2.00\\times 10^{-4}$    \\\\\n  \\bottomrule[2pt]\n\n\\end{tabular}\n\n\\end{center}\nNote.---Col.~(1): the initial NS mass. Col.~(2): the initial donor mass. Col.~(3): the initial orbital period. Col.~(4): the age of the donor star at the onset of RLOF. Col.~(5): the final mass of the NS. Col.~(6): the final mass of the WD. Col.~(7): final orbital period. Col.~(8): the type of the WD. Col.~(9): the mass accreted by the NS. Col.~(10): the duration of the mass transfer. Col.~(11): the maximum mass transfer rate.\n\\end{table}\n\n\\subsection{Further constraint from spin evolution}\n\n\\begin{figure}\n\\centering\n\\subfigure{\\includegraphics[scale=0.5]{f4a.eps}}\n\\subfigure{\\includegraphics[scale=0.5]{f4b.eps}}\n\\subfigure{\\includegraphics[scale=0.5]{f4c.eps}}\n\\subfigure{\\includegraphics[scale=0.5]{f4d.eps}}\n\\caption{Evolution of an IMXB with the initial NS mass, donor mass and orbital period $M_{1}=2.2M_{\\odot}$, $M_{2}=3.0M_{\\odot}$, and $P_{\\rm orb}=12.3$ d, respectively. The left panels show the evolution of $P_{\\rm orb}$, $M_1$, and $\\dot{M}$ as a function of the donor mass, and the right panels  the evolution of $\\dot{M}$ and chemical composition for the donor star as a function of time.\n\\label{figure3}}\n\n\\end{figure}\n\nNext we examine whether the mass transfer can accelerate the NS's spin period to milliseconds. The spin-up rate of the NS in a LMXB is determined by the rate of angular momentum transfer due to mass accretion,\n\\begin{equation}\n2\\pi I\\dot{P}\/P^2=\\dot{M}_1(GM_1R_1)^{1\/2},\n\\end{equation}\nwhere $P$ and $\\dot{P}$ are the spin period and its derivative of the NS respectively, and $R_1$ is the radius of the NS. Here we have assumed that the NS's magnetic field has been decayed so much that the accretion disk can extend to the surface of the NS. The amount of the accreted mass to produce a MSP can be roughly estimated to be\n\\begin{equation}\n\\Delta M\\sim 0.1 M_\\sun (M_1\/2M_\\sun)^{1\/2}(R_1\/10^6\\,{\\rm cm})^{-1\/2}(P\/3\\,{\\rm ms})^{-1}.\n\\end{equation}\nThe actual value of $\\Delta M$ could be smaller by a factor of $\\sim 2$ than that in Eq.~(13) if considering the effect of the NS magnetic field - accretion disk interaction. Combing this with Table 1, we notice that evolutions with Population II compositions seem to be more preferred for the formation of PSR J1640+2224.\n\nFig.~4 shows an example of the Population II evolutionary paths for PSR J1640+2224, which are able to reproduce the observed parameters of PSR J1640+2224. The evolution of this X-ray binary starts with a $2.2M_{\\odot}$ NS and a $3.0M_{\\odot}$ donor star in a 12.3 day orbit. The left panels show the evolution of $P_{\\rm orb}$, $M_1$, and $\\dot{M}_2$ as a function of the donor mass, and the right panels the evolution of $\\dot{M}_2$ and the chemical composition of the donor star. At the age of $t=236$ Myr  the donor star overflows its RL and transfers mass to the NS at a rate well above the Eddington-limit accretion rate.  Accordingly, the mass transfer lasts $\\sim$ 1.1 Myr, and the donor loses $\\sim 3.1\\,M_{\\odot}$ mass. Due to the extensive mass loss, the orbital period does not decrease significantly. Then the donor star shrinks and becomes detached from its RL. The next mass transfer phase lasts $\\sim$ 3.0 Myr with a  rate $\\sim 10^{-7}-10^{-6}M_{\\odot}$ yr$^{-1}$. After the mass transfer,  the NS has accreted material of $\\sim 0.062M_{\\odot}$.\nThe endpoint of the evolution is a binary consisting a recycled pulsar and a CO WD with a mass of $0.616M_{\\odot}$.\n\n\n\n\\section{DISCUSSION AND CONCLUSIONS}\n\nThe observed properties of PSR J1640+2224, such as its short spin period (3.16 ms), high WD mass ($M_{2}\\geq 0.4M_{\\odot}$ with $>90\\%$ confidence), long orbital period (175 days), and nearly circular orbit ($e=7.9725\\times 10^{-4}$) make it distinct from other MSPs and imply something unusual in its evolutionary history.\n\nIn this work, we have attempted to explore which evolutionary channels can reproduce the observed parameters of PSR J1640+2224. To achieve this goal, we consider different NS mass, donor mass and metal abundance ($Z=0.02$ and 0.001). Our mainly results are presented in Fig.~3 and Tables 1-5, and can be summarized as follows.\n\n1. For Population I chemical compositions, when $M_1\\simeq 2.0 \\,M_{\\odot}$, it is possible to simultaneously account for the WD mass ($\\sim 0.4 \\,M_{\\odot}$) and the orbital period ($\\sim$ 175 days). But if the WD mass $> 0.6 \\,M_{\\odot}$, the NS mass should be larger than $2.2 \\, M_{\\odot}$, and the donor star must be initially of intermediate-mass.\n\n2. For Population II chemical compositions, the evolution of original LMXBs containing a $1.4M_{\\odot}$ NS and a $1.0M_{\\odot}$ donor star may form PSR J1640+2224-like binaries with a $0.4M_{\\odot}$ WD companion. However, if the WD indeed has a mass higher than $0.6M_{\\odot}$, then the initial NS mass should be no less than $2.0M_{\\odot}$, and the initial donor  mass should be higher than $3.0 \\,M_{\\odot}$.\n\n3. When the NS spin evolution is taken into account, the evolutions with Population II compositions seem to be more preferred for the formation of PSR J1640+2224.\n\nAlmost all of our results predict a NS of initial mass higher than $2.0\\,M_{\\odot}$ (see Table 1). It is noted that \\citet{Fo16} demonstrated that J1640+2224 is likely a massive NS ($4.4^{+2.9}_{-2.0}\\,M_{\\odot}$) by analyzing the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) 9-year data set, although there are large uncertainties on the pulsar mass.\n\nThe birth masses of NSs depend on the physical mechanisms in core-collapse supernova explosions of massive stars, which are still highly uncertain \\citep{T96,J12}. Theoretical studies do claim that NSs could be born massive. For example, by simulating neutrino-powered supernova explosions in spherical symmetry, \\citet{U12} found that the birth masses of NSs vary within a range of $1.2-2.0M_{\\odot}$. \\citet{PT15} also numerically investigated the possible scope of the birth masses of NSs, and found that they could be as high as $1.9M_{\\odot}$. \\citet{S16} presented supernova simulations for a grid of massive stars and obtained the birth masses of NSs ranging from $1.2\\,M_{\\odot}$ to $1.8\\,M_{\\odot}$.\n\nObservationally, super-massive NSs have been discovered in quite a few X-ray binaries and binary MSPs \\citep{A16}. However, since they have experienced mass transfer processes, their current masses may not properly reflect their birth masses. For NSs in high-mass X-ray binaries (HMXBs), because they are relatively young (with an age $\\lesssim 10^7$ yr) and the efficiency of wind accretion is very low, their masses should be very close to those at birth \\citep{S12}. In HMXBs, the maximum NS mass measured is $2.12\\pm 0.16\\,M_{\\odot}$ for Vela X-1 \\citep{R11,F15}. For binary MSPs, the masses of super-massive NSs span between $\\sim 2.0\\,M_\\sun$ and $\\sim 2.9\\,M_\\sun$ \\citep[][for a recent review]{L19}. For example, PSR J0740+6620 is one of the most massive NSs ever accurately measured, with a mass $2.14^{+0.10}_{-0.09}\\,M_{\\odot}$ \\citep{C19}, a spin period of 2.88 ms and a He WD companion of mass $0.26M_{\\odot}$. These parameters mean that PSR J0740+6620 should have experienced extensive mass accretion during the LMXB phase, so its current mass may not represent its birth mass. However, recent studies suggested that mass transfer in LMXBs is likely to be highly non-conservative even at sub-Eddington mass transfer rates \\citep[e.g.,][]{T11,L11,A13,F16}.\nOur results show that, even with Eddington-limited accretion, a super-massive newborn NS is still required, at least for PSR J1640+2224.\n\nWe need to mention that our results are subject to several uncertainties, such as the treatment of convection in the donor star and the accretion efficiency during the mass transfer process. The convective overshooting parameter plays an important role in our work. Up to the present time two standard methods are used in modelling overshooting in numerical calculation. The first one is based on a simple extension of the the convectively mixed region above the boundary defined by the Schwarzschild criterion. This extension $l_{\\rm ov}$ is parameterized in terms of the local pressure scale height $H_{\\rm P}$ at the boundary,\n\\begin{equation}\nl_{\\rm ov}=\\alpha H_{\\rm P},\n\\end{equation}\nwhere $\\alpha$ is the convective overshooting parameter. In this model, various attempts have been undertaken to constrain the value of $\\alpha$. \\citet{Sc97} derived it to be 0.25 and 0.32 for stars in the mass range of $(2.5-7) \\,M_{\\odot}$ for eclipsing binary stars. \\citet{Ri00} and \\citet{Cl07} found that the value of $\\alpha$ is in the range of $0.1-0.6$, and the amount of overshooting increases systematically with the stellar mass. \\citet{Sa12} used asteroseismic analysis and obtained $\\alpha=0.1-0.3$ for $\\beta$ Cephei stars (with mass $M \\geq 8\\,M_{\\odot}$). \\citet{De10} analyzed the period space of SPB star HD 50230 ($M=7-8M_{\\odot}$) and suggested that the overshooting extent of the convective core is about $(0.2-0.3)\\,H_{\\rm P}$. \\citet{P14} analyzed the period spacings of KIC 10526294 ($M=3.25M_{\\odot}$) and suggested that $\\alpha$ is less than or equal to 0.15.\n\n\n\n\nIn an alternative approach, convective overshooting is considered to be a diffusive process with a diffusion parameter\n\\begin{equation}\nD=D_{0}\\exp(\\frac{-2z}{f_{\\rm ov}H_{\\rm P}})\n\\end{equation}\nwhere $z$ is the radial distance from the formal Schwarzschild border and $f_{\\rm ov}$ is a free parameter, $D_{0}$ is set as the scale of diffusive speed and derived from the convective velocity obtained from the mixing-length theory and taken below the Schwarzschild boundary \\citep{H00}.\n\n\\citet{H00} investigated the evolution of AGB stars with convective overshooting and set the diffusive convective parameter $f_{\\rm ov}=0.016$, which is widely adopted. \\citet{Mo15} analyzed KIC 10526294 with both the step function overshooting and exponentially decreasing overshooting, and found that the latter is better than the former for interpreting the observations with $f_{\\rm ov}=0.017-0.018$. \\citet{Mo16} obtained $f_{\\rm ov}=0.024\\pm 0.001$ for KIC 7760680. Based on the $k-\\omega$ model, \\citet{Guo19} concluded that $f_{\\rm ov}$ is about 0.008 for stars in the mass range of $(1.0-1.8)\\,M_{\\odot}$.\n\n\nIn Table 6, we compare the calculated results for I\/LMXB evolution with different overshooting parameters ($\\alpha=0.2$, $0.335$ and $f_{\\rm ov}=0.016$). The initial NS and donor masses are taken to be $M_{1}=2.0\\,M_{\\odot}$ and $M_{2}=2.5\\,M_{\\odot}$, respectively. From the results in Table 6, we conclude that the overshooting model with $\\alpha=0.2$ is nearly same as the diffusive overshooting model with $f_{\\rm ov}=0.016$, and that increasing the value of $\\alpha$ to 0.335 would result in more massive WDs and shorter orbital periods with the same initial parameters.\n\nIn Fig.~5, we compare the details of the evolution ($R_{2}, \\dot{M}_{2}, P_{\\rm orb}, M_{1}$ and $M_{2}$) for an IMXB with different overshooting parameters. The initial orbit period is set to be 5.35 days.  The evolutionary tracks with $\\alpha=0.2$ and $f_{\\rm ov}=0.016$ are nearly identical, while for $\\alpha=0.335$ RLOF starts $\\sim$ 60 Myr later than the others (panels (a) and (b)). In general, larger overshooting parameter produces a more massive core, less mass loss and smaller accreted mass by the NS (panel (d)).\n\nIn summary, PSR J1640+2224 was likely born massive, and this conclusion seems not sensitively dependent on the treatment of convective overshooting with reasonable parameters.\n\n\\begin{figure}[htbp]\n\n\\centerline{\\includegraphics[scale=0.6]{f5.eps}}\n\\caption{Comparison of the IMXB evolution with different overshoot parameters. The binary initially consists of a $2.0M_{\\odot}$ NS and a $2.5M_{\\odot}$ donor star, with an initial orbit period of 5.35 days. The top panels show the evolutions of $R_{2}$ and $\\dot{M}_{2}$ as a function of time, and the bottom panels the evolutions of $P_{\\rm orb}$ and stellar mass as a function of time. The red, blue and green curves represent the results with $\\alpha =0.2$, $0.335$, and $f_{\\rm ov}=0.016$, respectively. In panel (d), the solid and dashed lines denote the donor mass and the NS mass, respectively.\n\\label{figure7}}\n\n\\end{figure}\n\n\\acknowledgements\nWe are grateful the referee for helpful comments. This work was supported by the National Key Research and Development Program of China (2016YFA0400803), the Natural Science Foundation of China under grant No. 11773015 and Project U1838201 supported by NSFC and CAS.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe aim of this work and its forthcoming companions is to study Hodge theoretic aspects of tropical and non-archimedean geometry.\n\n\\medskip\n\nTropical geometry studies degenerations of algebraic varieties by enriching the theory of semistable models and their dual complexes by polyhedral geometry. This enrichment motivates the development of algebraic geometry for combinatorial and polyhedral spaces~\\cites{Mik06, MS, GS11, MZ, IKMZ, Cart, JSS} and their hybrid counter-parts, which contain a mixture of algebraic and polyhedral components~\\cites{Berkovich, AB15, BJ, HybridModuli, Mik06}.\n\n\\medskip\n\nIn dimension one, it is now well-understood that graphs and metric graphs are in many ways similar to Riemann surfaces and algebraic curves. For example, the analog of many classical theorems governing the geometry of Riemann surfaces, Riemann-Roch, Abel-Jacobi, Clifford and Torelli, are now established for graphs and metric graphs~\\cites{BN07, GK08, MZ08, Coppens, CV10, AB15}.\n\nThis viewpoint and its subsequent extensions and refinements have been quite powerful in applications in the past decade, and have resulted in several advances ranging from the study of curves and their moduli spaces (e.g. Brill-Noether theory~\\cite{CDPR}, maximum rank conjecture~\\cite{JP}, the Kodaira dimension of the moduli spaces of curves~\\cite{FJP}) to arithmetic geometry of curves (e.g. uniform bounds on the number of rational points on curves~\\cite{KRZ} and equidistribution results~\\cite{Ami14}). An overview of some of these results can be found in the survey paper~\\cite{BJ16}.\n\n\\medskip\n\nIt is expected that higher dimensional combinatorial and polyhedral analogs of graphs and metric graphs should fit into the same picture and fundamental theorems of algebraic geometry should have their combinatorial and polyhedral analogs.\n\n\\medskip\n\nRepresentative examples of this fresco in higher dimension have emerged in the pioneering work of McMullen and Karu~\\cites{Mcmullen, Karu} on hard Lefschetz theorem for polytopes, and more recently, in connection with Hodge theory, in the work of Adiprasito-Huh-Katz~\\cite{AHK} in the development of Hodge theory for matroids, in the work of Babaee-Huh~\\cite{BH17} on Demailly's strong reformulation of the Hodge conjecture for currents, in the work of Elias and Williamson~\\cite{EW} on Hodge theory of Soergel bimodules and positivity of Kazhdan-Lustig polynomials, and in the work of Adiprasito-Bj\\\"oner~\\cite{AB} and Adiprasito~\\cite{Adi18} on weak and combinatorial Lefschetz theorems, respectively.\n\n\\medskip\n\nOur aim here is to show that cohomology of smooth projective tropical varieties verify the tropical analogs of three fundamental theorems which govern the cohomology of complex projective varieties: Hard Lefschetz theorem, Hodge-Riemann relations and monodromy-weight conjecture. As we will explain below, this provides a uniform treatment and a generalization of many of the above mentioned results within the framework of tropical geometry.\n\n\\medskip\n\nIn the rest of this introduction, we provide a brief discussion of the results presented in this article.\n\n\n\n\\subsection{Matroids} Central objects underlying our results in this paper are combinatorial structures called matroids. Discovered by Whitney~\\cite{Whitney} in his study on classification of graphs with the same cycle structure, and independently by van der Waerden~\\cite{vdw} and Nakasawa, matroids provide combinatorial axiomatization of central notions of linear algebra. In this way, they lead to a common generalization of other combinatorial structures such as graphs, simplicial complexes and hyperplane arrangements. They appear naturally in diverse fields of mathematics, e.g., in combinatorics and statistical physics~\\cites{Tutte, MirPer, GGW, SS14, Piquerez}, optimization theory~\\cite{schrijver}, topology~\\cites{Mac91, GM92}, and algebraic geometry~\\cites{Mnev, Laf, BB, BL,sturmfels}. In tropical geometry, they are combinatorial structures underlying the idea of maximal degenerations of algebraic varieties~\\cites{deligne-md, KS}, and in this way give rise to a tropical notion of smoothness. We refer to~\\cites{wilson, kung, Oxl11} for an introduction and a historical account of the theory of matroids.\n\n\n\n\\subsection{Bergman fans} To any matroid $\\Ma$ over a ground set $E$ one associates a fan $\\Sigma_\\Ma$ called the \\emph{Bergman fan of $\\Ma$} which lives in the real vector space $\\rquot{\\mathbb{R}^E}{\\mathbb{R} (1,\\dots, 1)}$ and which is rational with respect to the lattice $\\rquot{\\mathbb{Z}^E}{\\mathbb{Z}(1, \\dots, 1)}$. In the case the matroid is given by an arrangement of hyperplanes, the Bergman fan can be identified with the tropicalization of the complement of the hyperplane arrangement, for the coordinates given by the linear functions which define the hyperplane arrangement~\\cite{AK}.\n\nThe fan $\\Sigma_\\Ma$ has the following description. First, recall that a \\emph{flat} of $\\Ma$ is a set $F$ of elements of $E$ which is closed with respect to the linear dependency, i.e., an element $e\\in E$ which is linearly dependent to $F$ already belongs to $F$. A non-empty flat $F \\subsetneq E$  is called \\emph{proper}. The Bergman fan $\\Sigma_\\Ma$ consists of cones $\\sigma_\\mathscr{F}$ in correspondence with \\emph{flags of proper flats of $\\Ma$}\n\\[\\mathscr{F}\\colon F_1 \\subsetneq F_2 \\subsetneq \\dots \\subsetneq \\dots \\subsetneq F_k, \\qquad k \\in \\mathbb N \\cup \\{0\\}. \\]\nFor such a flag $\\mathscr{F}$, the corresponding cone $\\sigma_\\mathscr{F}$ is the one generated by the characteristic vectors $\\e_{F_1}, \\dots, \\e_{F_k}$ of $F_1, \\dots, F_k$ in $\\mathbb{R}^E \/ \\mathbb{R}(1, \\dots, 1)$.  In particular, the rays of $\\Sigma_\\Ma$ are in bijection with the proper flats $F$ of $\\Ma$.\n\n\\smallskip\nWe call a \\emph{Bergman support} or a \\emph{tropical linear space} any subspace of a real vector space which is isomorphic via a linear map of real vector spaces to the support of the Bergman fan of a simple matroid. The terminology comes from~\\cites{Bergman, Spe08}.\n\n\\smallskip\nBy a \\emph{Bergman fan} we mean any fan which is supported on a Bergman support. In particular, Bergman fan $\\Sigma_\\Ma$ of a matroid $\\Ma$ is an example of a Bergman fan.\nMoreover, with this terminology, any complete fan in a real vector space is a Bergman fan. Indeed, for each integer $d$, the real vector space $\\mathbb{R}^d$ is the support of the Bergman fan of the uniform matroid $U_{d+1}$ of rank $d+1$ on ground set $[d]:=\\{0, 1, \\dots, d\\}$. (In $U_d$ every subset of $[d]$ is an independent set.) Thus, Bergman fans generalize both complete fans and Bergman fans of matroids. In addition, products of Bergman fans remain Bergman.\n\n\\smallskip\nIn this paper we consider \\emph{unimodular Bergman fans} (although it should be possible to generalize with some extra effort our results to simplicial Bergman fans): these are Bergman fans $\\Sigma$ in a real vector space $V$ which are moreover rational with respect to a full rank sublattice $N$ of $V$ and unimodular (smooth in the language of toric geometry). This means any cone $\\sigma$ in $\\Sigma$ is simplicial, that is generated by $\\dims{\\sigma}$, the dimension of $\\sigma$,  rays. Moreover, denoting $\\varrho_1, \\dots, \\varrho_{\\dims{\\sigma}}$ the rays of $\\sigma$, each ray $\\varrho_i$ is rational, and in addition, if $\\e_{\\varrho_i}$ denotes the primitive vector of the ray $\\varrho_i$, the set of vectors $\\e_{\\varrho_1}, \\dots, \\e_{\\varrho_{\\dims\\sigma}}$ is part of a basis of the lattice $N$. The product of two unimodular Bergman fans remains a unimodular Bergman fan.\n\n\\smallskip\nUnimodular Bergman fans are building blocks for \\emph{smooth tropical varieties} considered in this paper.\n\n\n\n\\subsection{Chow rings of matroids and more general Bergman fans} To a given simple matroid $\\Ma$ over a ground set $E$, one associates its Chow ring $A^\\bul(\\Ma)$, which is a graded ring defined by generators and relations, as follows.\n\nA \\emph{flat} of $\\Ma$ is a set $F$ of elements of $E$ which is closed with respect to the linear dependency: i.e., an element $e\\in E$ which is linearly dependent to $F$ already belongs to $F$. To any flat $F$ of $\\Ma$ one associates a variable $\\x_F$. The Chow ring is then a quotient of the polynomial ring $\\mathbb Z[\\x_F \\,\\,: F \\textrm{ flat of }\\Ma]$ by the ideal $\\I_\\Ma$ generated by the polynomials\n\\begin{itemize}\n\\item $\\x_F \\x_H$ for any two \\emph{incomparable} flats $F$ and $H$, i.e., with $F \\not\\subset H$ and $H \\not\\subset F$;\n\\item $\\sum_{F \\ni e} \\x_F - \\sum_{F \\ni f} \\x_F$ for any pair of elements $e,f\\in E$.\n\\end{itemize}\n\nIn the case the matroid $\\Ma$ is given by an arrangement of hyperplanes, the Chow ring gets identified with the Chow ring of the wonderful compactification of the complement of the hyperplane arrangement~\\cites{CP95, FY}.\n\n\\medskip\n\nLet now $\\Sigma$ be any unimodular Bergman fan in a vector space $V$ and let $N$ be the corresponding full rank lattice of $V$, so $V = N_\\mathbb{R} :=N\\otimes\\mathbb{R} $.\n\n\\smallskip\nThe Chow ring of $\\Sigma$ denoted by $A^\\bul(\\Sigma)$ is defined as the quotient of the polynomial ring $\\mathbb Z[\\x_\\varrho \\mid \\varrho\\in \\Sigma_1]$, with generators $\\x_{\\varrho}$ associated to the rays $\\varrho$ of $\\Sigma$, by the ideal $\\I_1 +\\I_2$ defined as follows:\n\\begin{itemize}\n\\item $\\I_1$ is the ideal generated by all monomials of the form $\\prod_{\\varrho \\in S} \\x_\\varrho$ for any subset $S \\subseteq \\Sigma_1$ which do not form a cone in $\\Sigma$; and\n\\item $\\I_2$ is the ideal generated by the elements of the form\n\\[\\sum_{\\varrho\\in \\Sigma_1} \\langle m, \\e_{\\varrho}\\rangle \\x_\\varrho\\]\nfor any $m$ in the dual lattice $M = N^\\vee$,\n\\end{itemize}\nwhere $\\e_\\varrho$ denotes the primitive vector of $\\varrho$, and $\\langle \\,\\cdot\\,, \\cdot\\,\\rangle$ is the duality pairing between $M$ and $N$.\n\n\\smallskip\nThis ring is known to be the Chow ring of the toric variety $\\P_\\Sigma$ defined by $\\Sigma$, \\cf.~\\cites{Dan78, BDP90, Bri96, FS}.\n\n\\smallskip\nIn the case $\\Sigma=\\Sigma_\\Ma$ is the Bergman fan of a simple matroid $\\Ma$, by the description of the fan $\\Sigma_\\Ma$, it is easy to see that the ring $A^\\bul(\\Sigma_\\Ma)$ coincides with the Chow ring $A^\\bul(\\Ma)$ of the matroid, introduced previously.\n\n\n\n\\subsection{Canonical compactifications} The \\emph{canonical compactification} of a simplicial fan $\\Sigma$ in a real vector space $V \\simeq \\mathbb{R}^n$ is defined as the result of completing any cone of $\\Sigma$ to a \\emph{hypercube}, by adding some faces at \\emph{infinity}. This is defined more precisely as follows. The fan $\\Sigma$ can be used to define a partial compactification of $V$ which we denote by $\\TP_\\Sigma$. The space $\\TP_\\Sigma$ is the tropical analog of the toric variety associated to $\\Sigma$. In fact, in the case the fan $\\Sigma$ is rational with respect to a full rank lattice $N$ in $V$, the tropical toric variety $\\TP_\\Sigma$ coincides with the tropicalization of the toric variety $\\P_\\Sigma$ associated to $\\Sigma$, see e.g.~\\cites{Payne, Thuillier}. These partial compactifications of $V$ coincide as well with the ones called \\emph{manifolds with corners} considered in~\\cites{AMRT, Oda}.\\\\\nThe canonical compactification of $\\Sigma$ denoted by $\\comp \\Sigma$ is then defined as the closure of $\\Sigma$ in the tropical toric variety $\\TP_\\Sigma$. The space $\\comp\\Sigma$ has the following description as the \\emph{cubical completion} of $\\Sigma$, see Section~\\ref{sec:tropvar} for a more precise description. Any ray $\\varrho$ of $\\Sigma$ becomes a segment by adding a point $\\infty_\\varrho$ at infinity to $\\varrho$. Any two dimensional cone $\\sigma$ with two rays $\\varrho_1$ and $\\varrho_2$ becomes a parallelogram with vertex set the origin, $\\infty_{\\varrho_{1}}, \\infty_{\\varrho_2}$, and a new point $\\infty_\\sigma$ associated to $\\sigma$. The higher dimensional cones are completed similarly to parallelepipeds.\n\n\\medskip\n\nCanonical compactifications of fans serve as building blocks for the description and the study of more general tropical varieties, as we will explain below.\n\n\n\n\\subsection{Hodge isomorphism theorem and Poincar\\'e duality} \\label{sec:poincareduality}\nLet $\\Sigma$ be a unimodular rational Bergman fan and let $\\comp \\Sigma$ be the canonical compactification of $\\Sigma$.\nThe compactification $\\comp \\Sigma$ is a \\emph{smooth tropical variety} in the sense that it is locally modelled by unimodular rational Bergman fans, see Section~\\ref{sec:smoothness_intro} below and Section~\\ref{sec:tropvar} for a precise definition of smoothness. In this specific case, which concerns compactifications of Bergman fans, if the fan $\\Sigma$ has support given by a simple matroid $\\Ma$, which is not unique in general, the local fans associated to $\\comp \\Sigma$ have support a Bergman fan associated to \\emph{restrictions} of \\emph{quotients} of $\\Ma$, or their products.\n\n\\medskip\n\nDenote by $H^{p,q}_\\trop(\\comp \\Sigma)$ the \\emph{tropical cohomology group} of $\\comp \\Sigma$ of bidegree $(p,q)$ introduced by Itenberg-Katzarkov-Mikhalkin-Zharkov in~\\cite{IKMZ}. We recall the definition of these groups in Section~\\ref{sec:tropvar}. We prove in~\\cite{AP} the following general theorem which allows in particular, when applied to the Bergman fan $\\Sigma_\\Ma$ of a matroid $\\Ma$, to reinterpret the results of Adiprasito-Huh-Katz~\\cite{AHK} as statements about the cohomology groups of a specific projective tropical variety, the canonical compactification of the Bergman fan of the matroid.\n\n\\begin{thm}[Hodge isomorphism for Bergman fans] \\label{thm:HI} For any unimodular Bergman fan $\\Sigma$ of dimension $d$, and for any non-negative integer $p\\leq d$, there is an isomorphism\n\\[A^p(\\Sigma) \\xrightarrow{\\ \\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$\\ }} H^{p,p}_\\trop(\\comp \\Sigma, \\mathbb{Z}).\\]\nThese isomorphisms induce an isomorphism of $\\mathbb{Z}$-algebras between the Chow ring of $\\Sigma$ and the tropical cohomology ring $\\bigoplus_{p=0}^d H_\\trop^{p,p}(\\comp \\Sigma, \\mathbb{Z})$. Furthermore, the cohomology groups $H_\\trop^{p,q}(\\comp \\Sigma, \\mathbb{Z})$ for $p\\neq q$ are all trivial.\n\\end{thm}\n\nSaying it differently, the theorem states that the \\emph{cycle class map} from Chow groups to tropical cohomology is an isomorphism, and in particular, the tropical cohomology groups of $\\comp \\Sigma$ are generated by Hodge classes.\n\n\\smallskip\nThe theorem can be viewed as the tropical analog to a theorem of Feichtner and Yuzvinsky~\\cite{FY}, which proves a similar theorem for wonderful compactifications of the complement of a hyperplane arrangement.\n\n\\smallskip\nAs a corollary of the above theorem, we obtain the following result.\n\\begin{thm}[Poincar\\'e duality] \\label{thm:pd} Let $\\Sigma$ be a unimodular Bergman fan of dimension $d$. There is a canonical isomorphism\n$ \\deg\\colon A^{d}(\\Sigma)\\to\\mathbb{Z}$ called the degree map. With respect to this isomorphism, the Chow ring of $\\Sigma$ satisfies the Poincar\\'e duality, in the sense that the pairing\n\\[ \\begin{tikzcd}[row sep=tiny, column sep=scriptsize]\nA^k(\\Sigma) \\times A^{d-k}(\\Sigma) \\rar& A^{d}(\\Sigma) \\rar[\"\\deg\"', \"\\sim\"]& \\mathbb{Z}\\\\\n(a,b) \\arrow[mapsto]{rr} && \\deg(ab)\n\\end{tikzcd} \\]\nis non-degenerate.\n\\end{thm}\nThis is in fact a consequence of Theorem~\\ref{thm:HI}, the observation that the canonical compactification $\\comp \\Sigma$ is a smooth tropical variety, and the fact that smooth tropical varieties satisfy Poincar\\'e duality, established by Jell-Shaw-Smacka~\\cite{JSS} for rational coefficients and by Jell-Rau-Shaw~\\cite{JRS19} for integral coefficients.\n\n\\medskip\n\nWe note that Theorem~\\ref{thm:pd} is independently proved by Gross-Shokrieh~\\cite{GS}.\n\n\n\n\\subsection{Hodge-Riemann and Hard Lefschetz for Bergman fans} \\label{sec:local-intro} Theorem~\\ref{thm:pd} generalizes Poincar\\'e duality for matroids proved in~\\cite{AHK} to any unimodular Bergman fan. In Section~\\ref{sec:local} we will show that Chow rings of \\emph{quasi-projective} unimodular Bergman fans verify Hard Lefschetz theorem and Hodge-Riemann relations, thus generalizing results of Adiprasito-Huh-Katz~\\cite{AHK} to any quasi-projective unimodular Bergman fan. The results are as follows.\n\n\\smallskip\nFor a unimodular Bergman fan $\\Sigma$ of dimension $d$ and an element $\\ell \\in A^1(\\Sigma)$, we say that the pair $(\\Sigma, \\ell)$ verifies the \\emph{Hard Lefschetz property $\\HL(\\Sigma, \\ell)$} if the following holds:\n\n\\medskip\n\n\\noindent (Hard Lefschetz) for any non-negative integer $k \\leq \\frac d2$, the multiplication map by $\\ell^{d-2k}$ induces an isomorphism\n\\[\\begin{tikzcd}[column sep=large]\n A^k(\\Sigma) \\arrow{r}{\\sim}& A^{d-k}(\\Sigma)\\,\\,, \\qquad a \\arrow[mapsto]{r} & \\ell^{d-2k}\\cdot a.\n\\end{tikzcd}\n\\]\n\n\\medskip\n\nWe say the pair $(\\Sigma, \\ell)$ verifies \\emph{Hodge-Riemann relations $\\HR(\\Sigma, \\ell)$} if for any non-negative integer $k\\leq \\frac d2$, the following holds:\n\n\\medskip\n\n\\noindent (Hodge-Riemann relations) for any non-negative integer $k \\leq \\frac d2$, the symmetric bilinear form\n\\[\\begin{tikzcd}[column sep=large]\n A^k(\\Sigma) \\times A^{k}(\\Sigma) \\arrow{r}& \\mathbb{Z} \\,\\,, \\qquad (a,b) \\arrow[mapsto]{r} & (-1)^k\\deg(\\ell^{d-2k}\\cdot a\\cdot b)\n\\end{tikzcd}\n\\]\nis positive definite on the \\emph{primitive part} $P^k(\\Sigma) \\subseteq A^k(\\Sigma)$, which by definition, is the kernel of the multiplication by $\\ell^{d-2k+1}$ from $A^k(\\Sigma)$ to $A^{d-k+1}(\\Sigma)$.\n\\begin{thm}[K\\\"ahler package for Bergman fans]\\label{thm-intro:HL-HR-local} Let $\\Sigma$ be a quasi-projective unimodular Bergman fan of dimension $d$. For any ample element $\\ell\\in A^1(\\Sigma)$, the pair $(\\Sigma, \\ell)$ verifies $\\HL(\\Sigma, \\ell)$ and $\\HR(\\Sigma, \\ell)$.\n\\end{thm}\nHere $\\ell$ is called \\emph{ample} if $\\ell$ can be represented in the form $\\sum_{\\varrho \\in \\Sigma_1} f(\\e_\\varrho) \\x_\\varrho$ for a strictly convex cone-wise linear function $f$ on the fan $\\Sigma$, see Section~\\ref{sec:tropvar} for the definition of strict convexity. A fan which admits such a function is called \\emph{quasi-projective}. We note that the Bergman fan $\\Sigma_\\Ma$ of any matroid $\\Ma$ is quasi-projective.\n\n\\smallskip\nIn addition to providing a generalization of~\\cite{AHK} to any quasi-projective Bergman fan, our methods lead to arguably more transparent proofs of these results when restricted to Bergman fans of matroids treated in~\\cite{AHK}. An alternate proof of the main result of~\\cite{AHK} for the Chow ring of a matroid based on semi-simple decomposition of the Chow ring has been obtained independently by Braden-Huh-Matherne-Proudfoot-Wang~\\cite{BHMPW}, as well as in the recent independent and simultaneous work of Ardila-Denham-Huh~\\cite{ADH}.\n\n\\medskip\n\nThe main idea in our approach which allows to proceed by induction, basically by starting from trivial cases and deducing the theorem in full generality, are the \\emph{ascent} and \\emph{descent} properties proved in Section~\\ref{sec:local}. These properties are also the final piece of arguments we will need in the global setting to conclude the proof of the main theorems of this paper. The possibility of having these nice properties is guaranteed by Keel's lemma for fans in the local setting, and by our projective bundle formula for tropical cohomology groups in the global setting, \\cf. the discussion below and the corresponding sections for more details.\n\n\n\n\\subsection{Smoothness in tropical geometry} \\label{sec:smoothness_intro} A tropical variety $\\mathfrak X$ is an \\emph{extended polyhedral space} with a choice of an affine structure. A polyhedral complex structure on a tropical variety is a polyhedral complex structure which induces the affine structure. We refer to Section~\\ref{sec:tropvar} which provides the precise definitions of these terminologies.\n\n\\smallskip\nA tropical variety is called \\emph{smooth} if it can be locally modeled by supports of Bergman fans \\cites{IKMZ, MZ, JSS}. This means, locally, around any of its points, the variety looks like the support of a Bergman fan.\n\n\\smallskip\nThe smoothness is meant to reflect in polyhedral geometry the idea of \\emph{maximal degeneracy} for varieties defined over non-Archimedean fields. Examples of smooth tropical varieties include smooth affine manifolds, canonical compactifications of Bergman fans, and tropicalizations of Mumford curves~\\cite{Jell-MC}.\n\n\\smallskip\nMore generally, to get a good notion of smoothness in tropical geometry, one would need to fix, as in differential topology, a good class of fans and their supports which can serve as local charts. One such class which emerges from the results in our work and the work of Deligne~\\cite{deligne-md} is that of fans which satisfy the Poincar\\'e duality for their tropical cohomology groups and which have canonical compactifications with vanishing $H^{p,q}_\\trop$ cohomology groups for all $p\\neq q$. By Hodge isomorphism Theorem~\\ref{thm:HI} and Poincar\\'e duality~\\cites{JSS, JRS19}, this class contains all Bergman fans, including therefore complete fans, and it would be certainly interesting to have a complete characterization of this class.\n\n\n\n\\subsection{Hodge-Riemann and Hard Lefschetz for smooth projective tropical varieties} In view of Hodge isomorphism Theorem~\\ref{thm:HI} and K\\\"ahler package for Bergman fans, Theorem~\\ref{thm-intro:HL-HR-local}, the results in Section~\\ref{sec:local-intro} can be regarded as statements concerning the tropical cohomology ring of the canonical compactifications $\\comp \\Sigma$ of unimodular Bergman fans. Moreover, one can show that when $\\Sigma$ is quasi-projective, the canonical compactification is projective~\\cite{AP-geom}. One of our aim in this paper is to show that all the results stated above hold more generally for the tropical cohomology groups of any smooth projective tropical variety, and more generally, for K\\\"ahler tropical varieties, a concept we introduce in a moment.\n\n\\medskip\n\nLet $V =N_\\mathbb{R}$ be a vector space of dimension $n$ and $N$ a full rank sublattice in $V.$ For a fan $\\Delta$ in $V$, we denote by $\\TP_{\\Delta}$ the corresponding \\emph{tropical toric variety}, which is a partial compactification of $V$.\n\nLet now $\\mathscr Y$ be a polyhedral subspace of $V$, and let $Y$ be a unimodular polyhedral complex structure on $\\mathscr Y$ with a quasi-projective unimodular \\emph{recession fan} $Y_\\infty$, \\cf. Section~\\ref{sec:tropvar} for the definitions of these terminologies. Let $\\mathfrak X$ be the closure of $\\mathscr Y$ in $\\TP_{Y_\\infty}$, equipped with the induced polyhedral structure that we denote by $X$. Let $\\ell$ be a strictly convex cone-wise linear function on $Y_\\infty$. We will show that $\\ell$ defines an element $\\omega$ of $H_\\trop^{1,1}(\\TP_{Y_\\infty}, \\mathbb{R})$, which lives in $H_\\trop^{1,1}(\\TP_{Y_\\infty}, \\mathbb{Q})$ if $\\ell$ has integral slopes. By restriction, this element gives an element in $H_\\trop^{1,1}(\\mathfrak X, \\mathbb{Q})$ which, by an abuse of the notation, we still denote by $\\omega$. Multiplication by $\\omega$ defines a Lefschetz operator on $H_\\trop^{\\bul, \\bul}(\\mathfrak X,\\mathbb{Q})$ still denoted by $\\omega$.\n\n\\medskip\n\nHere is the main theorem of this paper.\n\n\\begin{thm}[K\\\"ahler package for smooth projective tropical varieties]\\label{thm:main2} Notations as above and working with $\\mathbb{Q}$ coefficients, we have\n\\begin{itemize}\n\\item \\emph{(weight-monodromy conjecture)} For any pair of non-negative integers $p\\geq q$, we have an isomorphism\n\\[\\phi\\colon H_{\\trop}^{p,q}(\\mathfrak X) \\xrightarrow{\\sim} H_{\\trop}^{q,p}(\\mathfrak X).\\]\n\n\\item \\emph{(Hard Lefschetz)} For $p+q \\leq d\/2$, the Lefschetz operator $\\ell$ induces an isomorphism\n\\[\\ell^{d- p-q}\\colon H_{\\trop}^{p,q}(\\mathfrak X) \\to H^{d-q, d-p}_{\\trop}(\\mathfrak X).\\]\n\n\\item \\emph{(Hodge-Riemann)} The pairing $\\bigl< \\,\\cdot\\, , \\ell^{d-p-q} \\phi\\,\\cdot\\, \\bigr>$ induces a positive-definite symmetric bilinear form on the primitive part $P^{p,q}$ of $H_{\\trop}^{p,q}$, where $\\bigl< \\,\\cdot\\,,\\cdot\\, \\bigr>$ denotes the pairing\n\\begin{equation*}\nH^{p,q}_\\trop(\\mathfrak X) \\otimes H_\\trop^{d-p,d-q}(\\mathfrak X) \\longrightarrow H_\\trop^{d,d}(\\mathfrak X) \\xrightarrow[\\raisebox{3pt}{$\\sim$}]{\\quad\\deg\\quad} \\mathbb{Q}, \\qquad\n \\bigl< a,b \\bigr> \\longmapsto  (-1)^p\\deg(a \\cup b),\n\\end{equation*}\nand $\\phi$ is the isomorphism given in the first point.\n\\end{itemize}\n\\end{thm}\n\nThe first part of the theorem answers in positive a conjecture of Mikhalkin and Zharkov from~\\cite{MZ} where they prove the isomorphism of $H^{p,q}_\\trop(\\mathfrak X)$ and $H^{q,p}_\\trop(\\mathfrak X)$ in the \\emph{approximable case}. Recall that a projective tropical variety $\\mathfrak X$ is called \\emph{approximable} if it is the \\emph{tropical limit} of  a family of complex projective manifolds.\n\n\\medskip\n\nVarious versions of weak Lefschetz and vanishing theorems for projective tropical varieties were previously obtained by Adiprasito and Bj\\\"oner in~\\cite{AB}. Our work answers several questions raised in their work, \\cf.~\\cite{AB}*{Section 11}, as well as~\\cite{AB}*{Section 9}, and the results presented in the next sections.\n\n\\smallskip\nWe mention that an integral version of the weak Lefschetz theorem for hypersurfaces was obtained recently by Arnal-Renaudineau-Shaw in~\\cite{ARS}.\n\n\n\n\\subsection{Hodge index theorem for tropical surfaces} Our main theorem in the case of smooth tropical surfaces implies the following tropical version of the Hodge index theorem. Let $\\mathfrak X$ be a smooth projective tropical surface. By Poincar\\'e duality, we have $H^{2,2}_\\trop(\\mathfrak X) \\simeq \\mathbb{Q}$, and the cup-product on cohomology leads to the pairing\n\\[\\bigl< \\,\\cdot\\, , \\cdot\\, \\bigr> \\colon H^{1,1}_\\trop(\\mathfrak X) \\times H^{1,1}_\\trop(\\mathfrak X) \\to \\mathbb{Q},\\]\nwhich coincides with the intersection pairing on tropical homology group $H_{1,1}(\\mathfrak X)$.\n\\begin{thm}[Hodge index theorem for tropical surfaces]\\label{thm:hodgeindex} The signature of the intersection pairing on $H^{1,1}_\\trop(\\mathfrak X)$ is equal to $(1+b_2, h^{1,1}- 1- b_2)$ where $b_2$ is the second Betti number of $\\mathfrak X$ and $h^{1,1} =  \\dim_\\mathbb{Q} H^{1,1}_\\trop(\\mathfrak X)$.\n\\end{thm}\nThis theorem was previously known  by explicit computations for an infinite family of surfaces in $\\TP^3$, that of \\emph{floor decomposed surfaces of varying degree $d$} in $\\TP^3$, see~\\cite{Sha13b}. In that case, the second Betti number counts the number of interior points in the standard simplex of width $d$, i.e., with vertices $(0,0,0), (d,0,0), (0,d,0), (0,0,d)$.\n\n\\medskip\n\n\\subsection{} What follows next gives an overview of the materials we need to establish in order to  conclude the proof of Theorem~\\ref{thm:main2}. A sketch of our strategy appears in Section~\\ref{sec:skecth_intro} and could be consulted at this point ahead of going through the details of the next coming sections.\n\n\\subsection{Tropical K\\\"ahler forms and classes} \\label{sec:kahler-intro} The proof of the theorems above are all based on the notion of \\emph{K\\\"ahler forms} and their corresponding \\emph{K\\\"ahler classes} in tropical geometry that we introduce in this paper. Our definition is motivated by the study of the tropical Steenbrink sequence and its Hodge-Lefschetz structure, see below for a succinct presentation. A closely related notion of K\\\"ahler classes in non-Archimedean geometry has been introduced and studied in an unpublished work of Kontsevich-Tschinkel~\\cite{KT} and in the work of Yu on non-Archimedean Gromov compactness~\\cite{Yu} (see also~\\cites{Bou, BFJ15, BGS, CLD, GK17, Zhang} for related work).\n\n\\medskip\n\nConsider a smooth compact tropical variety $\\mathfrak X$ equipped with a unimodular polyhedral complex structure $X$. The polyhedral structure is naturally stratified according to \\emph{sedentarity} of its faces, which is a measure of \\emph{how far and in which direction at infinity} a point of $\\mathfrak X$ lies. We denote by $X_\\f$ the \\emph{finite part} of $X$ consisting of all the faces which do not touch the \\emph{part at infinity}, \\cf. Section~\\ref{sec:tropvar} for the precise definition. A \\emph{K\u00e4hler form} for $X$ is the data of ample classes $\\ell^v$ in the Chow ring of the local fan around the vertex $v$, for each vertex $v\\in X_\\f$, such that for each edge $e=\\{u,v\\}$ of $X_\\f$, the restriction of the two classes $\\ell^v$ and $\\ell^u$ in the Chow ring of the local fan around the edge $e$ are equal.\n\nWe show in Theorem~\\ref{thm:KahlerClass} that any K\\\"ahler form defines a class in $H_\\trop^{1,1}(\\mathfrak X)$. We call any class $\\omega$ induced by a K\\\"ahler form coming from a unimodular triangulation a \\emph{K\\\"ahler class}. A smooth compact tropical variety is then called \\emph{K\\\"ahler} if it admits a K\u00e4hler class. In particular, our definition of K\\\"ahler requires the variety to admit a unimodular triangulation. We note however that it is possible to remedy this by working with tropical Dolbeault cohomology.\n\n\\smallskip\nAs in the classical setting, we prove the following theorem.\n\\begin{thm} A smooth projective tropical variety which admits a quasi-projective triangulation is K\\\"ahler.\n\\end{thm}\nWe believe that \\emph{all smooth projective tropical varieties admit quasi-projective triangulations}, and plan to come back to this question in a separate publication. In the next section, we explain a weaker triangulation theorem we will prove in this paper which will be enough for our purpose.\n\n\n\n\\subsection{Quasi-projective and unimodular triangulations of polyhedral spaces} Our treatment of projective tropical varieties is based on the existence of what we call \\emph{regular unimodular triangulations} of rational polyhedral spaces in $\\mathbb{R}^n$. Regular triangulations are fundamental in the study of polytopes and their applications across different fields in mathematics and computer science, \\cf. the book by De Lorea-Rambau-Santos~\\cite{DRS} and the book by Gelfand-Kapranov-Zelvinsky~\\cite{GKZ}, one of the pioneers to the field of tropical geometry, for the definition and its relevance in algebraic geometry.\n\n\\smallskip\nRegularity for triangulations of a polytope is a notion of convexity, and as such, can be defined for any polyhedral subspace of $\\mathbb{R}^n$. Generalizing the pioneering result of Kemp-Knudson-Mumford and Saint-Donat~\\cite{KKMS} in the proof of the semistable reduction theorem, as well as previous variants proved by Itenberg-Kazarkov-Mikhalkin-Zharkov in~\\cite{IKMZ} and W{\\l}odarczyk~\\cite{Wlo97}, we obtain the following theorem on the existence of unimodular  quasi-projective triangulations.\n\n\\begin{thm}[Triangulation theorem]\\label{thm:regulartriangulations} Let $X$ be a rational polyhedral complex in $\\mathbb{R}^n$. There exists a regular triangulation of $X$ which is quasi-projective and unimodular with respect to the lattice $\\frac 1k \\mathbb{Z}^n$, for some integer $k\\in \\mathbb N$.\n\\end{thm}\n\nWe expect stronger versions of the theorem, as well as a theory of \\emph{secondary structures on regular triangulations}, and plan to come back to these questions in a future work.\n\n\n\n\\subsection{Canonical compactifications of polyhedral spaces}\nLet $\\mathscr Y$ be a polyhedral space in $V = N_\\mathbb{R}$. The \\emph{asymptotic cone} of $\\mathscr Y$ which we denote by $\\mathscr Y_\\infty$ is defined as the pointwise limit of the rescaled subsets $\\mathfrak X\/t$ when $t$ goes to $+\\infty$. The term asymptotic cone is borrowed from geometric group theory, originating from the pioneering work of Gromov~\\cites{Grom81, Grom83}, and exceptionally refers here to non-necessarily convex cones.\n\n\\medskip\n\nLet $\\mathscr Y$ be a smooth polyhedral space in $V = N_\\mathbb{R}$. Let $\\Delta$ be a unimodular fan structure on the asymptotic cone of $\\mathscr Y_\\infty$. Let $\\mathfrak X$ be the closure of $\\mathscr Y$ in $\\TP_\\Delta$. The compactification $\\mathfrak X$ is then smooth. We call it the \\emph{canonical compactification of $\\mathscr Y$ with respect to $\\Delta$}.\n\n\\smallskip\nAs in the theory of toric varieties, the tropical toric variety $\\TP_\\Delta$ has a natural stratification into  \\emph{tropical toric orbits}. For each cone $\\eta \\in \\Delta$, we have in particular the corresponding tropical toric subvariety denoted by $\\TP_\\Delta^\\eta$ and defined as the closure of the torus orbit associated to $\\eta$.\n\n\\medskip\n\nFor the canonical compactification $\\mathfrak X$ of $\\mathscr Y$ as above, we define for any $\\eta \\in \\Delta$ the closed stratum $D^\\eta$ of $\\mathfrak X$ as the intersection $\\mathfrak X \\cap \\TP_\\Delta^\\eta$. With these definitions, we see that the compactification boundary $D := \\mathfrak X \\setminus \\mathscr Y$ is a \\emph{simple normal crossing divisor} in $\\mathfrak X$, meaning that\n\\begin{itemize}\n\\item $D^{\\conezero} = \\mathfrak X$, where $\\conezero$ is the zero cone in $V$ consisting of the origin.\n\\item $D^\\eta$ are all smooth of dimension $d-\\dims{\\eta}$.\n\\item We have \\[D^\\eta = \\bigcap_{\\substack{\\varrho \\prec \\eta \\\\ \\dims{\\varrho}=1}} D^\\varrho.\\]\n\\end{itemize}\n\n\n\n\\subsection{Projective bundle formula} Notations as in the previous section, let $\\delta$ be a cone in $\\Delta$, and denote by $\\Delta'$ the fan obtained by the barycentric star subdivision of $\\Delta$. Denote by $\\mathfrak X'$ the closure of $\\mathscr Y$ in $\\TP_{\\Delta'}$ and $\\pi\\colon \\mathfrak X' \\to \\mathfrak X$ the projection map. The tropical variety $\\mathfrak X'$ is smooth again.\n\n\\smallskip\nLet $\\rho$ be the new ray in $\\Delta'$ obtained after the star subdivision of $\\delta \\in \\Delta$.  Consider the corresponding \\emph{tropical divisor} $D'^\\rho \\subseteq \\mathfrak X'$, i.e., the closed stratum in $\\mathfrak X'$ associated to $\\rho$.\n\n\\medskip\n\nThe divisor $D'^\\rho$ defines by the tropical cycle class map an element $\\class(D'^\\rho)$ of $H^{1,1}_\\trop(\\mathfrak X')$.\n\n\\smallskip\nFor any smooth compact tropical variety $\\mathscr W$, and for each integer $k$, we define the $k$-th cohomology of $\\mathscr W$ by\n\\[H^k(\\mathscr W) := \\sum_{p+q=k} H_\\trop^{p,q}(\\mathscr W).\\]\n\n\\begin{thm}[Projective bundle formula] \\label{thm:keelglobal-intro}\nWe have an isomorphism\n\\[H^k(\\mathfrak X' ) \\simeq H^k(\\mathfrak X) \\oplus T H^{k-2}(D^\\delta) \\oplus T^2 H^{k-4}(D^\\delta) \\oplus \\dots \\oplus T^{\\dims{\\delta}-1}H^{k-2\\dims{\\delta}+2}(D^\\delta).\\]\n\\end{thm}\nHere the map from the right hand side to the left hand side restricts to $\\pi^*$ on $H^k(\\mathfrak X)$ and sends $T$ to $-\\class(D'^\\rho)$. It is given on each factor $T^s H^{k-2s}(D^\\delta)$ by\n\\[ \\begin{array}{rcl}\nT^s H^{k-2s}(D^\\delta) & \\longrightarrow & H^k(\\mathfrak X') \\\\[1em]\nT^s \\alpha & \\longmapsto & (-1)^s \\class(D'^\\rho)^{s-1} \\cup \\pi^*\\circ \\gys(\\alpha),\n\\end{array} \\]\nwhere $\\pi^*$ and $\\gys$ are the \\emph{pull-back} and \\emph{Gysin} maps for the tropical cohomology groups, with respect to the projection $\\pi\\colon \\mathfrak X' \\to \\mathfrak X$ and the inclusion $D^\\delta \\hookrightarrow \\mathfrak X$, respectively.\n\n\\smallskip\nThe decomposition stated in the theorem provides for each pair of non-negative integers $(p,q)$,  a decomposition of the form\n\\[H^{p,q}(\\mathfrak X') \\simeq H^{p,q}(\\mathfrak X) \\oplus T H^{p-1, q-1}(D^\\delta) \\oplus T^2 H^{p-2, q-2}(D^\\delta) \\oplus \\dots \\oplus T^{\\dims{\\delta}-1}H^{p-\\dims\\delta+1,q - \\dims\\delta+1}(D^\\delta).\\]\n\n\\begin{remark}[Tropical Chern classes] The theorem describes the cohomology of the \\emph{blow-up} $\\mathfrak X'$ of $\\mathfrak X$ along the subvariety $D^\\delta \\subset \\mathfrak X$. Our proof actually provides more generally a projective bundle theorem for the projective bundle associated to a vector bundle $E$ on a smooth tropical variety $\\mathfrak X$. As in the classical setting, it allows to define Chern classes of vector bundles in tropical geometry over smooth projective tropical varieties. \\\\\nIn the situation above, we get\n\\[T^{\\dims{\\delta}} + c_1(N_{D^\\delta})T^{\\dims\\delta-1} +c_2(N_{D^\\delta})T^{\\dims\\delta-2} +\\dots + c_{\\dims \\delta}(N_{D^\\delta}) =0 \\]\nwhere $c_i$ are the Chern classes of the \\emph{normal bundle} $N_{D^\\delta}$ in $\\mathfrak X$.\\\\\nChern classes of matroids were previously defined and studied by L{\\'o}pez de Medrano, Rinc{\\'o}n, and Shaw in~\\cite{LRS}.\n\\end{remark}\n\n\n\n\\subsection{Tropical Steenbrink sequence} Let $\\mathfrak X$ be a smooth compact tropical variety and let $X$ be a unimodular polyhedral structure on $\\mathfrak X$. Denote by $X_\\f$ the set of faces of $X$ whose closures do not intersect the boundary at infinity of $\\mathfrak X$, i.e., the set of compact faces of $X_\\conezero$.\n\n\\medskip\n\nInspired by the shape of the first page of the Steenbrink spectral sequence~\\cite{Ste76}, we define bigraded groups $\\ST_1^{a,b}$ with a collection of maps between them as follows.\n\n\\medskip\n\nFor all pair of integers $a, b \\in \\mathbb Z$, we define\n\\[ \\ST_1^{a,b} := \\bigoplus_{s \\geq \\abs a \\\\ s \\equiv a \\pmod 2} \\ST_1^{a,b,s} \\]\nwhere\n\\[ \\ST_1^{a,b,s} = \\bigoplus_{\\delta \\in X_\\f \\\\ \\dims\\delta =s} H^{a+b-s}(\\comp \\Sigma^\\delta). \\]\nHere $\\Sigma^\\delta$ is the \\emph{star fan} of $X$ around $\\delta$ (also called the \\emph{transversal fan} of $\\delta$ in the literature), and $\\comp \\Sigma^\\delta = \\comp{\\Sigma^\\delta}$ is the canonical compactification of $\\Sigma^\\delta$.\n\n\\medskip\n\nThe bigraded groups $\\ST_1^{a,b}$ come with a collection of maps\n\\[\\i^{a,b\\,*} \\colon \\ST^{a,b}_1 \\to \\ST_1^{a+1, b} \\qquad \\textrm{and} \\qquad \\gys^{a,b}\\colon \\ST^{a,b}_1 \\to \\ST_1^{a+1, b}.\\]\nBoth these maps are defined by our sign convention introduced in Section~\\ref{sec:steenbrink} as an alternating sum of the corresponding maps on the level of cohomology groups appearing in the definition of $\\ST_1^{a,b,s}$ above. In practice, we drop the indices and denote simply by $\\i^*$ and $\\gys$ the corresponding maps.\n\n\\medskip\n\nUsing these two maps, we define the \\emph{differential} $\\d\\colon \\ST_1^{a,b} \\to \\ST_1^{a+1,b}$ as the sum $\\d = \\i^*+ \\gys$. For a unimodular triangulation $X$ of $\\mathfrak X$ and for any integer $b$, we will show that the differential $\\d$ makes $\\ST_1^{\\bul,b}$ into a cochain complex.\n\n\\smallskip\nFor a cochain complex $(C^\\bul, \\d)$, denote by $H^a(C^\\bul, \\d)$ its $a$-th cohomology group, i.e.,\n\\[H^a(C^\\bul, \\d) = \\frac{\\ker\\Bigl(\\d\\colon \\, C^{a} \\rightarrow C^{a+1}\\Bigr)}{\\Im\\Bigl(\\d\\colon \\, C^{a-1} \\rightarrow C^{a}\\Bigr)}. \\]\n\nWe prove the following comparison theorem.\n\\begin{thm}[Steenbrink-Tropical comparison theorem] \\label{thm:steenbrink-intro}\nThe cohomology of $(\\ST_1^{\\bul,b},\\d)$ is described as follows. For $b$ odd, all the terms $\\ST_1^{a,b}$ are zero, and the cohomology is vanishing. For $b$ even, let $b=2p$ for $p \\in \\mathbb Z$. Then for $q\\in \\mathbb Z$, we have a canonical isomorphism\n\\[H^{q-p}(\\ST^{\\bul,2p}_1, \\d) \\simeq H^{p,q}_{\\trop}(\\mathfrak X).\\]\n\\end{thm}\n\n\\medskip\n\nIn the approximable case, i.e., when $\\mathfrak X$ arises as the tropicalization of a family of complex projective varieties, this theorem was proved by Itenberg-Katzarkov-Mikhalkin-Zharkov in~\\cite{IKMZ}; see also the work of Gross-Siebert~\\cites{GS10, GS06} for a similar statement for the special case where $\\mathfrak X$ is an integral affine manifold with singularities.\n\n\\medskip\n\nOur proof is inspired by the one given in the approximable case~\\cites{IKMZ}, as well as by the use of the sheaf of logarithmic differentials on both Deligne's construction of a mixed Hodge structure on the cohomology of a smooth algebraic variety~\\cite{Deligne-Hodge2} and Steenbrink's construction of the limit mixed Hodge structure on the special fiber of a semistable degeneration~\\cite{Ste76}.\n\n\\medskip\n\nIn order to prove the theorem in this generality, we will introduce and develop a certain number of tools which we hope could be of independent interest and which will be used in our forthcoming work.\n\n\\smallskip\nIn particular, we will prove first an analogous result in the tropical setting of the \\emph{Deligne resolution} which gives a resolution of the coefficient groups $\\SF^p$ with cohomology groups of the canonically compactified fans $\\comp \\Sigma^\\delta$. (The coefficient group $\\SF^p$ is the discrete tropical analog of the sheaf of holomorphic forms of degree $p$, and is used to define the tropical cohomology groups $H^{p,q}(\\mathfrak X)$ for all non-negative integer $q$.) \\\\\nIn the approximable case, this resolution is a consequence of the Deligne spectral sequence in mixed Hodge theory as was observed in~\\cite{IKMZ}. The proof we present for the general setting is based on the hypercohomology of a complex of sheaves which provides a resolution of the \\emph{sheaf of tropical holomorphic forms} of a given order. It further uses Poincar\\'e duality as well as our Hodge isomorphism Theorem~\\ref{thm:HI}, which provides a description of the tropical cohomology groups of the canonically compactified fans.\n\n\\smallskip\nInspired by the weight filtration on the sheaf of logarithmic differentials, we define a natural weight filtration on the coefficient groups $\\SF^p(\\,\\cdot\\,)$, also somehow explicitly present in~\\cite{IKMZ},  and study the corresponding spectral sequence on the level of tropical cohomology groups. The resolution of the coefficient groups given by the Deligne exact sequence gives a double complex which allows to calculate the cohomology of the graded cochain complex associated to the weight filtration.\n\n\\smallskip\nWe then show that the spectral sequence associated to the double complex corresponding to the weight filtration which abuts to the tropical cohomology groups, abuts as well to the cohomology groups of the Steenbrink cochain complex.  The proof of this latter fact is based on an \\emph{unfolding of the Steenbrink sequence into a double complex}, which allows to define new Steenbrink type spectral sequences on each degree, and a \\emph{spectral resolution lemma} which allows to make a bridge between spectral sequences.\n\n\\medskip\n\nWhile the treatment we give of these constructions might appear quite technical, we would like to note that this should be merely regarded as a manifestation of the rich combinatorial structure of the Steenbrink spectral sequence, and the geometric informations it contains. We hope the effort made to make a \\emph{surgery} of this spectral sequence in this paper should paid off in further applications and developments, some of them will appear in our forthcoming work.\n\n\n\n\\subsection{Main theorem for triangulated tropical K\\\"ahler varieties}\nIn order to prove Theorem~\\ref{thm:main2}, we first establish it for a unimodular triangulation of a smooth tropical variety which admits a K\\\"ahler form. The advantage of these triangulations is that we can use the tropical Steenbrink spectral sequence in order to study the tropical cohomology groups. We show that this spectral sequence can be endowed with a \\emph{Hodge-Lefschetz structure}, where the monodromy operator is the analog of the one arising in the classical Steenbrink spectral sequence and the Lefschetz operator corresponds to multiplication with the K\\\"ahler form. This leads to the following theorem.\n\n\\begin{thm}\\label{thm:main1-intro} Let $X$ be a unimodular triangulation of a smooth tropical variety which admits a K\\\"ahler form given by classes $\\ell^v \\in H^{1,1}(v)$ for $v$ vertices of the triangulation. Denote by $\\ell$ the corresponding Lefschetz operator and by $N$ the corresponding monodromy operator. We have\n\\begin{itemize}\n\\item \\emph{(Weight-monodromy conjecture)} For $q>p$ two non-negative integers, we get an isomorphism\n\\[N^{q-p} \\colon H_{\\trop}^{q,p}(X) \\to H_{\\trop}^{p,q}(X).\\]\n\n\\item \\emph{(Hard Lefschetz)} For $p+q \\leq d\/2$, the Lefschetz operator $\\ell$ induces an isomorphism\n\\[\\ell^{d- p-q}\\colon H_{\\trop}^{p,q}(X) \\to H^{d-q, d-p}_{\\trop}(X).\\]\n\n\\item \\emph{(Hodge-Riemann)} The pairing $(-1)^p \\bigl< \\,\\cdot\\,, \\ell^{d-p-q} N^{q-p}\\,\\cdot\\,\\bigr>$ induces a positive-definite pairing on the primitive part $P^{q,p}$, where $\\bigl< \\,\\cdot\\,, \\cdot\\, \\bigr>$ is the natural pairing\n\n\\[\\bigl< \\,\\cdot\\, , \\cdot\\,\\bigr> \\colon H^{q,p}(X) \\otimes H^{d-q,d-p}(X) \\to H^{d,d}(X) \\simeq \\mathbb Q.\\]\n\\end{itemize}\n\\end{thm}\n\n\n\n\\subsection{Hodge-Lefschetz structures} In order to prove the theorem of the previous section, we show that $\\ST_1^{\\bul, \\bul}$ and the tropical cohomology $H^{\\bul,\\bul}_\\trop$ admit a Hodge-Lefschetz structure. To define this, we need to introduce the monodromy and Lefschetz operators on $\\ST_1^{\\bul, \\bul}$.\n\nThe monodromy operator is of bidegree $(2,-2)$ and it is defined in such a way to mimic the one obtained from the Steenbrink spectral sequence if we had a semistable family of complex varieties.\nSo it is the data of maps $N^{a,b} \\colon \\ST_1^{a,b} \\to \\ST_1^{a+2, b-2}$ defined as\n\\[N^{a,b} = \\bigoplus_{s \\geq \\abs a \\\\ s \\equiv a \\mod 2} N^{a,b,s}\\]\nwith\n\\begin{equation}\nN^{a,b, s} = \\begin{cases} \\mathrm{id} \\colon\\ST_1^{a,b,s} \\to \\ST_1^{a+2,b-2,s}   & \\textrm{if } s \\geq \\abs{a+2}, \\\\\n0 & \\textrm{otherwise}.\n\\end{cases}\n\\end{equation}\nMoreover, it coincides with the monodromy operator on the level of Dolbeault cohomology groups defined in~\\cite{Liu19}, via our Steenbrink-Tropical comparison Theorem~\\ref{thm:steenbrink-intro} and the comparison theorem between Dolbeault and tropical cohomology groups proved in~\\cite{JSS}.\n\n\\medskip\n\nThe definition of the Lefschetz operator depends on the choice of the K\\\"ahler form $\\ell$. Recall that a K\\\"ahler form is the data of ample classes $\\ell^v$ at vertices $v$ which are moreover compatible along edges. This guaranties the compatibility of restrictions to higher dimensional faces giving an ample element $\\ell^\\delta \\in H^2(\\delta) = A^1(\\delta)$ for all faces $\\delta \\in X_\\f$.\n\n\\medskip\n\nThe Lefschetz operator $\\ell^{a,b} \\colon \\ST_1^{a,b} \\to \\ST_1^{a, b+2} $ is then defined as the sum \\[\\ell^{a,b} = \\bigoplus \\ell^{a,b,s} \\colon \\bigoplus_{s \\geq \\abs{a} \\\\\ns \\equiv a \\pmod 2} \\ST_1^{a,b, s} \\to \\bigoplus_{s \\geq \\abs{a} \\\\\ns \\equiv a \\pmod 2} \\ST_1^{a,b+2, s}\\]\nwhere $\\ell^{a,b,s}$ is the sum of the contributions of local ample elements\n\\[\\ell^{a,b,s} = \\bigoplus \\ell^\\delta \\colon \\bigoplus_{\\delta \\in X_\\f \\\\ \\dims{\\delta} =s} H^{a+b-s}(\\delta) \\longrightarrow    \\bigoplus_{\\delta \\in X_\\f \\\\ \\dims{\\delta} =s} H^{a+b+2-s}(\\delta).\\]\nIn practice, the indices are dropped and this is denoted simply  by $\\ell$.\n\n\\smallskip\nThe Monodromy and Lefschetz operators $N$ and $\\ell$ verify the following properties\n\\begin{itemize}\n\\item $[\\ell, N] = 0$, $[\\ell, \\i^*] = [\\ell, \\gys] =0$, and $[N,\\i^*] = [N, \\gys] =0$.\n\n\\smallskip\n\\item for a pair of integers $a,b$ with $a+b \\geq d$, the map\n\\[\\ell^{d-a-b} = \\underbrace{\\ell \\circ \\ell \\circ \\dots \\circ \\ell}_{(d-a-b) \\,\\textrm{times}} \\colon \\ST_1^{a,b} \\longrightarrow \\ST_1^{a, 2d - 2a-b}\\]\nis an isomorphism.\n\n\\smallskip\n\\item for a pair of integers $a,b$ with $a\\leq 0$, the map\n\\[N^{-a} = \\underbrace{N \\circ N \\circ \\dots \\circ N}_{(-a) \\,\\textrm{times}} \\colon  \\ST_1^{a,b} \\longrightarrow \\ST_1^{-a, b + 2a}\\] is an isomorphism.\n\\end{itemize}\n\nThese properties show that the tropical Steenbrink spectral sequence with the two operators $\\ell$ and $N$ form a Hodge-Lefschetz structure. Moreover, we show that local polarizations allow to define a polarization on $\\ST_1^{\\bullet, \\bullet}$ leading to a polarized Hodge-Lefschetz structure.\nFrom this, we deduce that the cohomology of the tropical Steenbrink spectral sequence admits a polarized Hodge-Lefschetz structure. Using this, and combining the corresponding \\emph{primitive decomposition} with the comparison Theorem~\\ref{thm:steenbrink-intro}, we can finish the proof of Theorem~\\ref{thm:main1-intro}.\n\n\\medskip\n\nDifferential Hodge-Lefschetz structures are treated in Saito's work on Hodge modules~\\cite{Saito}, in the paper by Guill\\'en and Navarro Aznar on invariant cycle theorem~\\cite{GNA90} and in the upcoming book by Sabbah and Schnell~\\cite{SabSch} on mixed Hodge modules. We should however note that the tropical set-up is slightly different, in particular, the differential operator appearing in our setting is skew-symmetric with respect to the polarization. The proof we give of these results is elementary and does not make any recourse to representation theory, although it might be possible to recast in the language of representation theory the final combinatorial calculations we need to elaborate.\n\n\\medskip\n\nWe refer to Sections~\\ref{sec:kahler} and \\ref{sec:differential_HL_structure} for more details.\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:main2}}\\label{sec:skecth_intro}\nHaving explained all the needed ingredients, we now explain how the proof of the main theorem can be deduced. Starting with the projective tropical variety $\\mathfrak X$, and the quasi-projective unimodular fan structure $\\Delta$ induced on $\\mathfrak X_\\infty$ by the compactification we proceed as follows. The choice of the convex piecewise function $\\ell$ on $\\Delta$ gives an element $\\ell \\in H^{1,1}(\\mathfrak X)$. Using the ascent and descent properties, that we extend naturally from the local situation to the global setting using the projective bundle theorem, we show that it will be enough to treat the case where $\\mathfrak X$ admits a quasi-projective unimodular triangulation $X$ compatible with $\\Delta$.  In this case, we show that the class $\\ell \\in H^{1,1}(\\mathfrak X)$ is K\\\"ahler, and so we can apply Theorem~\\ref{thm:main1-intro} to conclude.\n\n\n\n\\subsection{Organization of the paper} The paper is organized as follows. In the next section, we provide the relevant background on tropical varieties, introduce canonical compactifications, and recall the definition of cohomology groups associated to them. This section introduces as well the corresponding terminology in polyhedral geometry we will be using all through the paper.\n\nIn Section~\\ref{sec:local} we present the generalization of the results of Adiprasito-Huh-Katz to any unimodular Bergman fan, thus proving Theorem~\\ref{thm-intro:HL-HR-local}. These local results are the basis for all the materials which will appear in the upcoming sections. Moreover, the ascent and descent properties used in the proof of local Hodge-Riemann relations will be again crucial in our proof of the global Hodge-Riemann relations.\n\nSection~\\ref{sec:triangulation}, which is somehow independent of the rest of the paper, is devoted to the proof of our triangulation theorem, cf. Theorem~\\ref{thm:regulartriangulations}. The results are crucial in that they allow to introduce a tropical analog to the Steenbrink spectral sequence, which will be the basis for all the results which come thereby after.\n\nIn Section~\\ref{sec:steenbrink} we study the tropical Steenbrink spectral sequence, introduce the weight filtration on tropical coefficient groups $\\SF^p$, and prove the tropical Deligne resolution theorem as well as the comparison theorem between Steenbrink sequence and tropical cohomology groups, cf. Theorem~\\ref{thm:steenbrink-intro}. For the ease of reading, few computational points of this section are treated separately in Appendix.\n\nK\\\"ahler tropical varieties are introduced in Section~\\ref{sec:kahler}. For a unimodular triangulation of a smooth tropical varieties which admits a K\\\"ahler form, the corresponding tropical Steenbrink spectral sequence can be endowed with a Hodge-Lefschetz structure, where the monodromy operator is the analog of the one arising in the classical Steenbrink spectral sequence and the Lefschetz operator corresponds to multiplication with the K\\\"ahler class. Using an adaptation of the theory of Hodge-Lefschetz structure to the tropical setting, Sections~\\ref{sec:kahler} and~\\ref{sec:differential_HL_structure} are devoted to the proof of the K\\\"ahler package for triangulated tropical varieties, cf. Theorem~\\ref{thm:main1-intro}. The fact that the Hodge-Lefchetz structure will be inherited in the cohomology in the tropical setting is treated in Section~\\ref{sec:differential_HL_structure}.\n\nThe last two sections are then devoted to the proof of Theorem~\\ref{thm:main2}. In Section~\\ref{sec:projective_bundle_theorem}, we  present the proof of the projective bundle theorem, cf. Theorem~\\ref{thm:keelglobal-intro}. Section~\\ref{sec:proofmaintheorem} then finishes the proof of the main theorem using the materials developed in the previous sections, ascent-descent property, projective bundle theorem and Theorem~\\ref{thm:main1-intro}.\n\n\\medskip\n\nThe results presented in Appendix show that a certain triple complex constructed in Section~\\ref{sec:steenbrink} provides a spectral resolution of the tropical spectral sequence, associated to the weight filtration on the tropical complex. Due to the calculatory nature of the content of this section, and necessity of introducing extra notations, we have decided to include it only as an appendix. It is also written somehow independently of the rest of the paper. This means we reproduce some of the materials from the paper there, and in few places, we adapt a terminology slightly different from the one used in the main body. We have found this more adapted both in terms of rigour and compactness to the purpose of this appendix which requires through case by case computations. Every time this change of terminology happens, we make a comment of comparison to the one used in the previous sections.\n\n\n\n\n\n\n\n\n\\section{Tropical varieties}\n\\label{sec:tropvar}\n\nThe aim of this section is to provide the necessary background on polyhedral spaces and tropical varieties. It contains a brief review of polyhedral geometry, matroids and their associated Bergman fans, tropical and Dolbeault cohomology, as well as a study of canonical compactifications of polyhedral spaces, which will be all used in the consequent sections. This will be also the occasion to fix the terminology and notations which will be used all through the text.\n\n\n\n\\subsection{Recollection in polyhedral geometry} \\label{sec:recol}\nA (closed convex) \\emph{polyhedron} of a real vector space $V\\simeq \\mathbb{R}^n$ is a non-empty intersection of a finite number of affine half-spaces. All through this paper, a polyhedron means a \\emph{strongly convex} one in the sense that we require it does not contain any affine line.\n\nLet $P$ be a polyhedron. The \\emph{affine tangent space} of $P$ denoted by $\\Tan{P}$ is by definition the smallest affine subspace of $\\mathbb{R}^n$ which contains $P$; it is precisely the set of all linear combinations of elements of $P$ whose coefficients sum up to one. The \\emph{linear tangent space}, or simply \\emph{tangent space}, of $P$ denoted by $\\TT P$ is the linear subspace of $\\mathbb{R}^n$ spanned by the differences $x-y$ for any pair of elements $x,y$ of $P$.\n\nThe \\emph{dimension} of $P$ is denoted by $\\dims{P}$, which is by definition that of its tangent space. A \\emph{face} of $P$ is either $P$ or any nom-empty intersection $P\\cap H$ for any affine hyperplane $H$ such that $P\\subset H^+$ where $H^+$ is one of the two half-space delimited by $H$. Note that a face of a polyhedron $P$ is itself a polyhedron. We use the notation $\\gamma\\prec\\delta$ for two polyhedra if $\\gamma$ is a face of $\\delta$. If moreover $\\gamma$ is of codimension one in $\\delta$, i.e., $\\dims\\gamma=\\dims\\delta-1$, then we write $\\gamma\\ssubface\\delta$. A face of dimension zero is called a \\emph{vertex} of $P$. A face of dimension one is called an \\emph{edge}. The \\emph{interior} of a polyhedron $P$ refers to the relative interior of $P$ in its affine tangent space, which is by definition the complement in $P$ of all the proper faces of $P$.\n\nA \\emph{cone} is a polyhedron with a unique vertex which is the origin of $\\mathbb{R}^n$. (By our assumption, we only consider strongly convex cones, which means that the cone does not include any line.) A compact polyhedron is called a \\emph{polytope}; equivalently, a polytope is a polyhedron which is the convex-hull of a finite number of points.\n\nSuppose now that the vector space $V \\simeq \\mathbb{R}^n$ comes with a full rank lattice $N \\simeq \\mathbb{Z}^n $. A polyhedron $P$ in $V$ is called \\emph{rational} if the half-spaces used to define $P$ can be defined in $N_\\mathbb{Q} \\simeq \\mathbb{Q}^n$. We denote by $N_P$ the full-rank lattice $N\\cap \\TT(P)$ in $\\TT(P)$.\n\nA polyhedron is called \\emph{integral with respect to $N$} if it is rational and all its vertices are in $N$. In the sequel, we simply omit to mention the underlying lattice $N$ if this is clear from the context.\n\nThe Minkowski sum of two polyhedra $P$ and $Q$ in $V$ is defined by\n\\[P+Q \\colon = \\bigl\\{x+ y \\st x\\in P, y \\in Q\\Bigr\\}.\\]\n\nBy Minkowski-Weyl theorem, every polyhedron $P$ can be written as the Minkowski sum $Q+\\sigma$ of a polytope $Q$ and a cone $\\sigma$. Moreover, one can choose $Q$ to be the convex-hull of the vertices of $P$ and the cone $\\sigma$ is uniquely characterized by the property that for any point $x\\in P$, $\\sigma$ is the maximal cone verifying $x+\\sigma\\subseteq P$. We denote this cone by $P_\\infty$.\n\nIt follows that for any polyhedron $P$, one can choose points $v_0,\\dots,v_k$ and vectors $u_1,\\dots,u_l$ in $V$ such that\n\\[ P=\\conv(v_0,\\dots,v_k)+\\sum_{i=1}^l\\mathbb{R}_+u_i. \\]\nHere and in the whole article, $\\mathbb{R}_+$ denotes the space of non-negative real numbers.\n\nA polyhedron $P$ is called \\emph{simplicial} if we can choose the points $v_i$ and the vectors $u_j$ in the above decomposition so that the collection of vectors $(v_1-v_0,\\dots,v_k-v_0,u_1,\\dots,u_l)$ be independent.\nIn this case, there is a unique choice for points $v_0,\\dots, v_k$, which will be the vertices of $P$. (And the vectors $u_1, \\dots, u_l$ are in bijection with the rays of $P_\\infty$.) We define\n\\[P_\\f:=\\conv(v_0,\\dots, v_k),\\]\nand we have $P = P_\\f + P_\\infty$. Furthermore, any point $x\\in P$ can be decomposed in a unique way as the sum $x=x_\\f+x_\\infty$ of a point $x_\\f\\in P_\\f$ and a point $x_\\infty\\in P_\\infty$.\n\nA simplicial polyhedron $P$ is called \\emph{unimodular with respect to the lattice} $N$ in $V$ if it is integral with respect to $N$ and moreover, the points $v_i$ and the vectors $u_j$ can be chosen in $N$ so that $(v_1-v_0,\\dots,v_k-v_0,u_1,\\dots,u_l)$ is part of a basis of $N$ (or equivalently, if they form a basis of the lattice $N_P = N\\cap\\TT(P)$).\n\n\\medskip\n\nA \\emph{polyhedral complex} in a real vector space $V \\simeq \\mathbb{R}^n$ is a finite non-empty set $X$ of polyhedra in $\\mathbb{R}^n$ called \\emph{faces of $X$} such that for any pair of faces $\\delta,\\delta'\\in X$\n\\begin{enumerate}[label={\\textnormal{(\\textit{\\roman*})}}]\n\\item \\label{enum:subface} any face of $\\delta$ belongs to $X$, and\n\\item the intersection $\\delta\\cap\\delta'$ is either empty or is a common face of $\\delta$ and $\\delta'$.\n\\end{enumerate}\nA finite collection of polyhedra $X$ which only verifies the first condition \\ref{enum:subface} is called a \\emph{polyhedral pseudo-complex}.\n\nIf $P$ is a polyhedron, the set of faces of $P$ form a polyhedral complex which we denote by $\\face{P}$. A polyhedral complex which has a unique vertex is called a \\enquote{\\emph{fan}} if the unique vertex is the origin of $V$. In this case, all the faces of the complex are cones. The \\emph{finite part of $X$} denoted by $X_\\f$ is the set of all compact faces of $X$, which is itself again a polyhedral complex. The \\emph{support of $X$} denoted by $\\supp{X}$ is the union of the faces of $X$ in $V$. A polyhedral complex whose support is the entire vector space $V$ is said to be \\emph{complete}.\n\nA polyhedral complex is called \\emph{simplicial}, \\emph{rational}, \\emph{integral}, or \\emph{unimodular} (with respect to the lattice $N$), if all faces of $X$ are simplicial, rational, integral, or unimodular (with respect to $N$), respectively.\n\nLet $X$ and $Y$ be two polyhedral complexes in a real vector space $V$. We say that $Y$ is a \\emph{subdivision} of $X$ if $X$ and $Y$ have the same support, and each face of $Y$ is included in a face of $X$. We say that $Y$ is a \\emph{triangulation} of $X$ if in addition $Y$ is simplicial. We say that $Y$ is a \\emph{subcomplex} of $X$ or that $X$ \\emph{contains} $Y$ if $Y \\subseteq X$. In particular, the support of $X$ contains that of $Y$. We say that $X$ \\emph{contains a subdivision of} $Y$ if there exists a subdivision $Z$ of $Y$ which is a subcomplex of $X$. The same terminology will be used for polyhedral pseudo-complexes.\n\n\\medskip\n\nLet $S$ be a subset of a real vector space $V \\simeq \\mathbb{R}^n$. The set of \\emph{affine linear functions on $S$} is defined as the restrictions  to $S$ of affine linear functions on $V$ and is denoted by $\\aff(S)$. For a polyhedral complex, we simplify the notation and write $\\aff(X)$ instead of $\\aff(\\supp X)$.\n\nLet $f\\colon \\supp{X} \\to \\mathbb{R}$ be a continuous function. We say that $f$ is \\emph{piecewise linear on $X$} if on each face $\\delta$ of $X$, the restriction $f\\rest \\delta$ of $f$ to $\\delta$ is affine linear. The set of piecewise linear functions on $X$ is denoted by $\\lpm(X)$. ($\\lpm$ reads \\emph{lin\\'eaire par morceaux}.)\n\nA function $f\\in\\lpm(X)$ is called \\emph{convex}, resp. \\emph{strictly convex}, if for each face $\\delta$ of $X$, there exists an affine linear function $\\ell \\in \\aff(V)$ such that $f-\\ell$ vanishes on $\\delta$ and is non-negative, resp. strictly positive, on $U \\setminus \\delta$ for an open neighborhood $U$ of the relative interior of $\\delta$ in $\\supp{X}$.\n\nWe denote by $\\K(X)$, resp.\\ , $\\K_+(X)$, the set of convex, resp. strictly convex, functions in $\\lpm(X)$. A polyhedral complex $X$ is called \\emph{quasi-projective} if the set $\\K_+(X)$ is non-empty. We will treat this notion of convexity in more detail in Section~\\ref{subsec:convexite}.\n\n\n\n\\subsection{Canonical compactifications of fans} \\label{sec:can-compact-fans-2}\n\nLet $\\eR = \\mathbb{R} \\cup \\{\\infty\\}$ be the extended real line with the topology induced by that of $\\mathbb{R}$ and a basis of open neighborhoods of infinity given by intervals $(a, \\infty]$ for $a\\in \\mathbb{R}$. Extending the addition of $\\mathbb{R}$ to $\\eR$ in a natural way, by setting $\\infty + a = \\infty$ for all $a \\in \\eR$, gives $\\eR$ the structure of a topological monoid. We call $(\\eR, +)$ the \\emph{monoid of tropical numbers} and denote by $\\eR_+ = \\mathbb{R}_+ \\cup\\{\\infty\\}$ the submonoid of non-negative tropical numbers with the induced topology.\n\nBoth monoids admit a natural scalar multiplication by non-negative real numbers (setting $0\\cdot\\infty=0$). Moreover, the multiplication by any factor is continuous. As such, $\\eR$ and $\\eR_+$ can be seen as module over the semiring $\\mathbb{R}_+$. Recall that modules of semirings are quite similar to classical modules over rings, except that such modules are commutative monoids instead of being abelian groups. Another important collection of  examples of topological modules over $\\mathbb{R}_+$ are the cones.\n\nWe can consider the tensor product of two modules over $\\mathbb{R}_+$. In this section, every tensor product will be over $\\mathbb{R}_+$, thus we will sometimes omit to mention it.\n\n\\medskip\n\nAll through, $N$ will be a free $\\mathbb{Z}$-module of finite rank and $M=N^{\\vee}$ denotes the dual of $N$. We denote by $N_\\mathbb{R}$ and $M_\\mathbb{R}$ the corresponding real vectors spaces, so we have $M_\\mathbb{R} =N_\\mathbb{R}^\\vee$. For a polyhedron $\\delta$ in $N_\\mathbb{R}$, we use both the notations $N_{\\delta, \\mathbb{R}}$ or $\\TT\\sigma$, depending on the context, to denote the \\emph{linear tangent space} to $\\delta$ which is the real vector subspace of $N_\\mathbb{R}$ generated by differences $x-y$ for pairs of elements $x, y$ in $\\delta$. Furthermore, we define the \\emph{normal vector space} of $\\delta$ denoted $N^{\\delta}_{\\mathbb{R}}$ by $N^{\\delta}_{\\mathbb{R}} := \\rquot{N_\\mathbb{R}}{N_{\\delta, \\mathbb{R}}}$. If the polyhedron $\\delta$ is rational, then we naturally get lattices of full rank $N^\\delta$ and $N_\\delta$ in $N^\\delta_\\mathbb{R}$ and $N_{\\delta, \\mathbb{R}}$, respectively.\n\n\\medskip\n\nBy convention, all through this article, we most of the time use $\\delta$ (or any other face) as a superscript to denote the quotient of some space by $\\TT\\delta$, or to denote elements related to this quotient. On the contrary, we use $\\delta$ as a subscript for subspaces of $\\TT\\delta$ or elements associated to these subspaces.\n\n\\medskip\n\nFor a fan $\\Sigma$ in $N_\\mathbb R$, we denote by $\\Sigma_k$ the set of $k$-dimensional cones of $\\Sigma$; elements of $\\Sigma_1$ are called \\emph{rays}. In the case the fan is rational, for any ray $\\varrho\\in\\Sigma_1$, we denote by $\\e_\\varrho$ the generator of $\\varrho\\cap N$.\n\n\\medskip\n\nFor any cone $\\sigma$, denote by $\\sigma^\\vee \\subseteq M_\\mathbb{R}$ the dual cone defined by\n\\[\\sigma^\\vee := \\Bigl\\{m \\in M_\\mathbb R \\st \\langle m, a \\rangle \\geq 0 \\ \\textrm{ for all }\\ a \\in \\sigma\\Bigr\\}. \\]\nWe also define the orthogonal plane to $\\sigma$ by\n\\[\\sigma^\\perp := \\Bigl\\{m \\in M_\\mathbb R \\st \\langle m, a \\rangle = 0 \\ \\textrm{ for all }\\ a \\in \\sigma\\Bigr\\}.  \\]\n\n\\medskip\n\nThe \\emph{canonical compactification} $\\comp \\sigma$ of a cone $\\sigma$, called as well the \\emph{extended cone} of $\\sigma$, is defined by\n\\[\\comp \\sigma := \\sigma \\otimes_{\\mathbb{R}_+} \\eR_+. \\]\nThe topology on $\\comp\\sigma$ is the finest one such that the endomorphisms\n\\[ z\\mapsto z+z', \\qquad z\\mapsto x\\otimes z \\quad\\text{ and }\\quad x\\mapsto z\\otimes a \\]\nare continuous for any $z'\\in\\comp\\sigma$, $x\\in\\sigma$ and $a\\in\\eR_+$.\nThe space $\\comp\\sigma$ is a compact topological space, whose restriction to $\\sigma$ coincides with the usual topology.\n\n\\begin{remark} \\label{rem:definition_dual}\nOne can also define $\\comp\\sigma$ in the following way. Let $\\tm$ be the category of topological monoids whose morphisms are continuous. Then\n\\[ \\comp\\sigma = \\Hom_\\tm(\\sigma^\\vee, \\eR_+). \\]\nThe advantage of this definition is that the link with compactification of toric varieties is more direct (cf. below). However, the description of the topology and of the general shape of $\\comp\\sigma$ is easier with the first definition.\n\\end{remark}\n\nThere is a distinguished point $\\infty_\\sigma$ in $\\comp \\sigma$ defined by $x\\otimes\\infty$ for any $x$ in the relative interior of $\\sigma$. The definition does not depend on the chosen $x$. Note that for the cone $\\conezero$, we have $\\infty_{\\conezero} = 0$.\n\nFor an inclusion of cones $\\tau \\prec \\sigma$, we get a map $\\comp \\tau \\subseteq \\comp \\sigma$. This inclusion identifies $\\comp \\tau$ as the topological closure of $\\tau$ in $\\comp \\sigma$.\n\n\\medskip\n\nLet $\\Sigma$ be a rational fan in $N_\\mathbb{R}$. The canonical compactification $\\comp \\Sigma$ is defined as the union of extended cones $\\comp \\sigma$ for any cone $\\sigma\\in\\Sigma$ where we identify $\\comp \\tau$ with the corresponding subspace of $\\comp \\sigma$ for any inclusion of cones $\\tau \\prec \\sigma$ in $\\Sigma$. The topology of $\\comp \\Sigma$ is the induced quotient topology. Each extended cone $\\comp \\sigma$ naturally embeds as a subspace of $\\comp \\Sigma$.\n\n\\medskip\n\nThe canonical compactification $\\comp \\Sigma$ of $\\Sigma$ naturally lives in a \\emph{partial compactification} of $N_\\mathbb{R}$ defined by $\\Sigma$ that we denote by $\\TP_\\Sigma$. We define $\\TP_\\Sigma$ as follows. For any cone $\\sigma$ in $\\Sigma$, we consider the space $\\~\\sigma=(\\sigma + \\sigma^\\perp) \\otimes_{\\mathbb{R}_+} \\eR$, endowed with the finest topology such that sum and the wedge product are continuous. We have a natural inclusion of $\\comp\\sigma$ into $\\~\\sigma$. Notice that $(\\sigma+\\sigma^\\perp)\\otimes\\mathbb{R}\\simeq N_\\mathbb{R}$. We set $N^\\sigma_{\\infty,\\mathbb{R}}:=\\infty_\\sigma + N_\\mathbb{R} \\subset \\~\\sigma$. Clearly $N^\\conezero_{\\infty,\\mathbb{R}}=N_\\mathbb{R}$. Moreover, from the fact that for any $x\\in\\sigma$ and any $\\lambda\\in\\mathbb{R}$, $\\infty_\\sigma + x\\otimes \\lambda=\\infty_\\sigma$, we can see the map\n\\[ N_\\mathbb{R} \\to N^\\sigma_{\\infty,\\mathbb{R}}, \\qquad z \\mapsto z+\\infty_\\sigma \\]\nas a projection along $\\sigma\\otimes\\mathbb{R}\\simeq N_{\\sigma,\\mathbb{R}}$. Indeed, one can prove that $N^\\sigma_{\\infty,\\mathbb{R}}\\simeq N^\\sigma_{\\mathbb{R}}$.\n\nThen, $\\~\\sigma$ is naturally stratified into a disjoint union of subspaces $N^\\tau_{\\infty,\\mathbb{R}}$, each isomorphic to $N^\\tau_\\mathbb{R}$, for $\\tau$ running over faces of $\\sigma$. Because of this isomorphism, we might sometimes omit the $\\infty$, when it is clear from the context that we are considering the stratum at infinity. Moreover, the inclusions $\\~\\tau\\subseteq\\~\\sigma$ for pairs of elements $\\tau \\prec \\sigma$ in $\\Sigma$ allow to glue these spaces and define the space $\\TP_\\Sigma$.\n\nThe partial compactification $\\TP_\\Sigma$ of $N_\\mathbb{R}$ is naturally stratified as the disjoint union of $N^\\sigma_{\\infty,\\mathbb{R}} \\simeq N^{\\sigma}_\\mathbb{R}$ for $\\sigma \\in \\Sigma$.\n\n\\begin{remark}[Remark \\ref{rem:definition_dual} continued] We can also define $\\~\\sigma$ as a set by\n\\[ \\~\\sigma = \\Hom_\\tm(\\sigma^\\vee, \\eR_+). \\]\nThis justifies the discussion of the next paragraph.\n\\end{remark}\n\nFor a rational fan $\\Sigma$, we have $\\TP_\\Sigma = \\mathrm{Trop}(\\P_\\Sigma)$ the extended tropicalization of the toric variety $\\P_\\Sigma$. In other words, $\\TP_\\Sigma$ can be viewed as the \\emph{tropical toric variety associated to the fan $\\Sigma$}. In particular, the tropical projective space $\\TP^n$ coincides with $\\TP_\\Sigma$ for $\\Sigma$ the standard fan of the projective space $\\P^n$. We refer to~\\cites{AP, BGJK, Kaj, Payne, Thuillier} for more details.\n\n\\medskip\n\nThe canonical compactification $\\comp\\Sigma$ of $\\Sigma$ admits a natural stratification into cones and fans that we describe now. Moreover, this stratification can be enriched into an \\emph{extended polyhedral structure}~\\cite{AP-geom}, see Section~\\ref{sec:extpc} for more details.\n\nConsider a cone $\\sigma \\in \\Sigma$ and a face $\\tau$ of $\\sigma$. Let $C^\\tau_\\sigma:=\\infty_\\tau+(\\sigma\\otimes 1) \\subseteq \\comp\\sigma$. From this description, one sees that $C^\\tau_\\sigma$ is isomorphic to the projection of the cone $\\sigma$ in the linear space $\\rquot{N_{\\sigma, \\mathbb R}}{N_{\\tau, \\mathbb R}} \\hookrightarrow N^{\\tau}_{\\infty,\\mathbb{R}}$. We denote by $\\C^\\tau_\\sigma$ the relative interior of $C^\\tau_\\sigma$.\n\nOne can show that the canonical compactification $\\comp \\Sigma$ is a disjoint union of (open) cones $\\C^\\tau_\\sigma$ for pairs $\\tau\\prec\\sigma$ of elements of $\\Sigma$.\n\nFor a fixed cone $\\tau \\in \\Sigma$, the collection of cones $C^\\tau_\\sigma$ for elements $\\sigma \\in \\Sigma$ with $\\tau \\prec \\sigma$ form a fan with origin $\\infty_\\tau$ that we denote by $\\Sigma_\\infty^\\tau\\subset{N^\\tau_{\\infty,\\mathbb{R}}}$. Note that with this terminology, for the cone $\\conezero$ of $\\Sigma$ we have $\\Sigma_\\infty^\\conezero = \\Sigma$.\n\nThe fan $\\Sigma_\\infty^\\tau$ is canonically isomorphic to the \\emph{star fan} of $\\tau$ in $\\Sigma$ denoted by $\\Sigma^\\tau$ and defined by\n\\[\\Sigma^\\tau := \\Bigl\\{\\, \\pi_\\tau (\\sigma) \\Bigst \\sigma \\succ \\tau \\textrm{ is a cone in $\\Sigma$} \\, \\Bigr\\}, \\]\nwhere $\\pi_\\tau\\colon N_{\\mathbb{R}} \\to N^\\tau_\\mathbb{R}$. Note that our use of the star fan is consistent with the one used in~\\cite{AHK} and differs from the usual terminology, e.g., in~\\cites{Karu, BBFK} where this is called \\emph{transversal fan}.\n\nIf there is no risk of confusion, we sometimes drop $\\infty$ and write $\\Sigma^\\tau$ when referring to the fan $\\Sigma_\\infty^\\tau$ based at $\\infty_\\tau$.\n\nFor any pair $(\\tau, \\sigma)$ with $\\tau \\prec \\sigma$, the closure $\\comp C^\\tau_\\sigma$ of $C^\\tau_\\sigma$ in $\\comp \\Sigma$ is the union of all the cones $\\C^{\\tau'}_{\\sigma'}$ with $\\tau \\prec \\tau' \\prec \\sigma' \\prec \\sigma$. The closure of $\\Sigma_\\infty^\\tau$ is indeed canonically isomorphic to the canonical compactification of $\\Sigma^\\tau \\subseteq N^{\\tau}_\\mathbb{R}$.\n\nThe stratification of $\\comp \\Sigma$ by cones $C^\\tau_\\sigma$ and their closures is called the \\emph{conical stratification} of $\\comp \\Sigma$. Note that there is a second stratification of $\\comp \\Sigma$ into fans $\\Sigma_\\infty^\\tau$ for $\\tau \\in \\Sigma$.\n\n\\medskip\n\nBy definition, from the inclusion of spaces $\\sigma\\otimes\\eR_+ \\simeq (\\sigma+\\sigma^\\perp)\\otimes\\eR_+ \\subset (\\sigma+\\sigma^\\perp)\\otimes\\eR$, we get an embedding $\\comp \\Sigma \\subseteq \\TP_\\Sigma$. Since the strata $\\Sigma^\\tau_\\infty$ of $\\comp \\Sigma$ lives in $N^\\tau_{\\infty, \\mathbb{R}}$, this embedding is compatible with the stratification of both spaces. In particular, $\\comp \\Sigma$ is the compactification of the fan $\\Sigma$ in the tropical toric variety $\\TP_\\Sigma$.\n\n\n\n\\subsection{Extended polyhedral structures}\\label{sec:extpc} We now describe an enrichment of the category of polyhedral complexes and polyhedral spaces into \\emph{extended} polyhedral complexes and \\emph{extended} polyhedral spaces by replacing the ambient spaces $V\\simeq \\mathbb{R}^n$ with $\\eR^n$, or more generally, with a partial compactification $\\TP_\\sigma$ for the fan $\\face\\sigma$ associated to a cone $\\sigma$ in $V$, as described in the previous section. For more details we refer to~\\cites{JSS, MZ, IKMZ}.\n\nLet $n$ be a natural number and let $x = (x_1, \\dots, x_n)$ be a point of $\\eR^n$. The \\emph{sedentarity} of $x$ denoted by $\\sed(x)$ is the set $J$ of all integers $j \\in [n]$ with $x_j = \\infty.$ The space $\\eR^n$ is stratified into subspaces $\\mathbb{R}^n_J$ for $J \\subseteq [n]$ where $\\mathbb{R}^n_J$ consists of all points of $x$ of sedentarity $J$. Note that $\\mathbb{R}^n_J \\simeq \\mathbb{R}^{n-\\card{J}}$.\n\nMore generally, let $\\sigma$ be a cone in $V$. By an abuse of the notation, we denote by $\\sigma$ the fan in $V$ which consists of $\\sigma$ and all its faces. By the construction described in the previous section, $\\sigma$ provides a partial compactification $\\TP_\\sigma$ of $V$, where we get a bijection between strata $V^\\tau_{\\infty}$ of $\\TP_\\sigma$ and faces $\\tau$ of the cone $\\sigma$. In the context of the previous section, the stratum $V^\\tau_\\infty$ was denoted $N^\\tau_{\\infty,\\mathbb{R}}$.\n\nGeneralizing the definition of the sedentarity, for any point $x\\in \\TP_\\sigma$ which lies in the stratum $V^\\tau_\\infty$, with $\\tau\\prec\\sigma$, the \\emph{sedentarity} of $x$ is by definition $\\sed(x):=\\tau$. For $\\sigma$ the positive quadrant in $\\mathbb{R}^n$, these definitions coincide with the ones in the preceding paragraph, as we get $\\TP_\\sigma  =\\eR^n$ and the subfaces of $\\sigma$ can be identified with the subsets $[n]$.\n\n\\medskip\n\nAn \\emph{extended polyhedron} $\\delta$ in $\\TP_\\sigma$ is by definition the topological closure in $\\TP_\\sigma$ of any polyhedron included in a strata $V^\\tau_\\infty$ for some $\\tau \\prec \\sigma$. A \\emph{face} of $\\delta$ is the topological closure of a face of $\\delta \\cap V^\\zeta_\\infty$ for some $\\zeta\\prec\\sigma$. An \\emph{extended polyhedral complex} in $\\TP_\\sigma$ is a finite collection $X$ of extended polyhedra in $\\TP_\\sigma$ such that the two following properties are verified.\n\\begin{itemize}\n\\item For any $\\delta \\in X$, any face $\\gamma$ of $\\delta$ is also an element of $X$.\n\\item For pair of elements $\\delta$ and $\\delta'$, the intersection $\\delta \\cap \\delta'$ is either empty or a common face of $\\delta$ and $\\delta'$.\n\\end{itemize}\nThe support of the extended polyhedral complex $\\supp X$ is by definition the union of $\\delta \\in X$. The space $\\mathfrak X = \\supp X$ is then called an \\emph{extended polyhedral subspace} of $\\TP_\\sigma$, and $X$ is called an \\emph{extended polyhedral structure} on $\\mathfrak X$.\n\n\\medskip\n\nWe now use extended polyhedral subspaces of partial compactifications of vector spaces of the form $\\TP_\\sigma$ as local charts to define more general extended polyhedral spaces.\n\nLet $n$ and $m$ be two natural numbers, and let $V_1\\simeq \\mathbb{R}^n$ and $V_2\\simeq \\mathbb{R}^m$ be two vector spaces with two cones $\\sigma_1$ and $\\sigma_2$ in $V_1$ and $V_2$, respectively. Let $\\phi\\colon V_1 \\to V_2$ be an affine map between the two spaces and denote by $A$ be the linear part of $\\phi$. Let $I$ be the set of rays $\\varrho\\prec\\sigma_1$ such that $A\\varrho$ lives inside $\\sigma_2$. Let $\\tau_I$ be the cone of $\\sigma_1$ which is generated by the rays in $I$. The affine map $\\phi$ can be then extended to a map $\\bigcup_{\\substack{\\zeta \\prec \\sigma_1 \\\\ \\zeta \\subseteq \\tau_I}} V^\\zeta_{1,\\infty} \\to \\TP_{\\sigma_2}$ denoted by $\\phi$ by an abuse of the notation. We call the extension an \\emph{extended affine map}. More generally, for an open subset $U \\subseteq \\TP_{\\sigma_1}$, a map $\\phi \\colon U \\to \\TP_{\\sigma_2}$ is called an \\emph{extended affine map} if it is the restriction to $U$ of an extended affine map $\\phi\\colon \\TP_{\\sigma_1} \\to \\TP_{\\sigma_2}$ as above. (The definition thus requires that $U$ is a subset of $\\bigcup_{\\substack{\\zeta \\prec \\sigma_1 \\\\ \\zeta \\subseteq \\tau_I}} V^\\zeta_{1,\\infty}$.) In the case $V_1$ and $V_2$ come with sublattices of full ranks, the extended affine map is called \\emph{integral} if the underlying map $\\phi$ is integral, i.e., if the linear part $A$ is integral with respect to the two lattices.\n\n\\medskip\n\nAn \\emph{extended polyhedral space} is a Hausdorff topological space $\\mathfrak X$ with a finite atlas of charts $\\Bigl(\\phi_i\\colon W_i \\to U_i \\subseteq \\mathfrak X_i\\Bigr)_{i\\in I}$ for $I$ a finite set such that\n\n\\begin{itemize}\n\\item $\\bigl\\{\\,W_i \\st {i\\in I}\\, \\bigr\\}$ form an open covering of $\\mathfrak X$;\n\\item $\\mathfrak X_i$ is an extended polyhedral subspace of $\\TP_{\\sigma_i}$ for a finite dimensional real vector space $V_i\\simeq\\mathbb{R}^{n_i}$ and a cone $\\sigma_i$ in $V_i$, and $U_i$ is an open subset of $\\mathfrak X_i$; and\n\\item $\\phi_i$ are homeomorphisms between $W_i$ and $U_i$ such that for any pair of indices $i,j\\in I$, the transition map\n\\[\\phi_j\\circ \\phi_i^{-1} \\colon \\phi_i(W_i \\cap W_j) \\to \\TP_{\\sigma_j}\\]\nis an extended affine map.\n\\end{itemize}\n\nThe extended polyhedral space is called \\emph{integral} if the transition maps are all integral with respect to the lattices $\\mathbb{Z}^{n_i}$ in $\\mathbb{R}^{n_i}$.\n\n\\medskip\n\nLet $\\mathfrak X$ be an extended polyhedral space with an atlas of charts $\\Bigl(\\,\\phi \\colon  W_i \\to U_i  \\subseteq \\mathfrak X_i\\Bigr)_{i\\in I}$ as above. A \\emph{face structure} on $\\mathfrak X$ is the choice of an extended polyhedral complex $X_i$ with $\\supp{X_i} = \\mathfrak X_i$ for each $i$ and a finite number of closed sets $\\theta_1, \\dots, \\theta_N$, for $N \\in \\mathbb N$, called \\emph{facets} which cover $\\mathfrak X$ such that the following two properties are verified.\n\n\\begin{itemize}\n\\item Each facet $\\theta_j$ is contained in some chart $W_i$ for $i\\in I$ such that $\\phi_i(\\theta_j)$ is the intersection of the open set $U_i$ with a face $\\eta_{j, i}$ of the polyhedral complex $X_i$.\n\\item For any subset $J \\subseteq [N]$, any $j \\in J$, and any chart $W_i$ which contains $\\theta_j$, the image by $\\phi_i$ of the intersection $\\bigcap_{j\\in J} \\theta_j$ in $U_i$ is the intersection of $U_i$ with a face of $\\eta_{j,i}$.\n\\end{itemize}\nA \\emph{face} of the face structure is the preimage by $\\phi_i$ of a face of $\\eta_{j,i}$ for any $j\\in[N]$ and for any $i\\in  I$ such that $\\theta_j\\subseteq W_i$. Each face is contained in a chart $W_i$  and the \\emph{sedentarity of a face $\\delta$ in the chart $W_i$} is defined as the sedentarity of any point in the relative interior of $\\phi_i(\\delta)$ seen in $X_i$.\n\n\\begin{prop} Let $\\Sigma$ be a fan in $N_\\mathbb{R}$. The canonical compactification $\\comp \\Sigma$ naturally admits the structure of an extended polyhedral space, and a face structure given by the closure $\\comp C^\\tau_\\sigma$ in $\\comp \\Sigma$ of the cone $C^\\tau_\\sigma$ in the conical stratification of $\\comp \\Sigma$, for pairs of cones $\\tau \\prec \\sigma$ in $\\Sigma$. The extended polyhedral structure is integral if $\\Sigma$ is a rational fan.\n\\end{prop}\n\\begin{proof} This is proved in~\\cite{AP-geom}.\n\\end{proof}\n\nNote that the sedentarity of the face $\\comp C^\\tau_\\sigma$ of the conical stratification of $\\comp \\Sigma$ is the set of rays of $\\tau$.\n\n\n\n\\subsection{Recession fan and canonical compactification of polyhedral complexes} \\label{sec:can-compact-polycomp} Let $X$ be a polyhedral complex in a real vector space $N_\\mathbb{R} \\simeq \\mathbb{R}^n$. The \\emph{recession pseudo-fan} of $X$ denoted by $X_\\infty$ is the set of cones $\\{\\delta_\\infty\\st\\delta\\in X\\}$. In fact the collection of cones $\\delta_\\infty$ do not necessary form a fan, as depicted in the following example; we will however see later in Section~\\ref{sec:triangulation} how to find a subdivision of $X$ whose recession pseudo-fan becomes a fan. In such a case, we will call $X_\\infty$ the \\emph{recession fan} of $X$. Recession fans are studied in~\\cite{BS11}.\n\n\\begin{example} Let $V = \\mathbb{R}^3$ and denote by $(e_1, e_2, e_3)$ the standard basis of $\\mathbb{R}^3$. Define\n\\begin{gather*}\n\\sigma_1=\\mathbb{R}_+e_1+\\mathbb{R}_+e_2\\subset\\mathbb{R}^3, \\\\\n\\sigma_2=\\mathbb{R}_+e_1+\\mathbb{R}_+(e_1+e_2)\\subset\\mathbb{R}^3, \\\\\nX=\\face{\\sigma_1}\\cup\\faceop\\left(e_3+\\sigma_2\\right).\n\\end{gather*}\nNote that $X_\\infty$ contains the cones $\\sigma_1$ and $\\sigma_2$ whose intersection is not a face of $\\sigma_1$.\n\\end{example}\n\nLet $X$ be a polyhedral complex in $N_\\mathbb{R} \\simeq \\mathbb{R}^n$ whose recession pseudo-fan $X_\\infty$ is a fan. Let $\\TP_{X_\\sminfty}$ be the corresponding tropical toric variety. The \\emph{canonical compactification of $X$} denoted by $\\comp X$ is by definition the closure of $X$ in $\\TP_{X_\\sminfty}$.\n\n\\medskip\n\nWe now describe a natural stratification of $\\comp X$. Let $\\sigma \\in X_\\infty$, and consider the corresponding stratum $N^{\\sigma}_{\\infty, \\mathbb{R}}$ of $\\TP_\\sigma$. Define $X^\\sigma_{\\infty}$ to be the intersection of $\\comp X$ with $N^{\\sigma}_{\\infty, \\mathbb{R}}$, that without risk of confusion, we simply denote $X^\\sigma$ by dropping $\\infty$. Note that $X^{\\conezero} = X$, which is the \\emph{open part} of the compactification $\\comp X$.\n\n\\medskip\n\nLet $D = \\comp X \\setminus X$ be the \\emph{boundary at infinity} of the canonical compactification of $X$. For each non-zero cone $\\sigma$ in $X_\\infty$, we denote by $D^\\sigma$ the closure of $X^\\sigma$ in $\\comp X$.\n\n\\begin{thm} [Tropical orbit-stratum correspondence]\\label{thm:orbit-stratum-correspondence}\nNotations as above, let $X$ be a polyhedral complex in $N_\\mathbb{R} \\simeq \\mathbb{R}^n$. We have the following.\n\\begin{enumerate}\n\\item For each cone $\\sigma \\in X_\\infty$, the corresponding strata $X^\\sigma$ is a polyhedral complex in $N^{\\sigma}_{\\infty, \\mathbb{R}}$.\n\\item The pseudo-recession fan $(X^{\\sigma})_\\infty$ of $X^\\sigma$ is a fan which coincides with the fan $(X_\\infty)_\\infty^{\\sigma}$ in $N_{\\infty, \\mathbb{R}}^\\sigma$ in the stratification of the canonical compactification $\\comp{X_\\infty}$ of the fan $X_\\infty$.\n\\item The closure $D^\\sigma$ of $X^\\sigma$ coincides with the canonical compactification of $X^{\\sigma}$ in $N^{\\sigma}_{\\infty, \\mathbb{R}}$, i.e., $D^\\sigma = \\comp{X^{\\sigma}}$.\n\\item If $X$ has pure dimension $d$, then each stratum $X^{\\sigma}$ as well as their closures $D^\\sigma$ have pure dimension $d -\\dims \\sigma$.\n\\end{enumerate}\nIn particular, the canonical compactification $\\comp X$ is an extended polyhedral structure with a face structure induced from that of $X$.\n\\end{thm}\n\n\\begin{proof} The theorem is proved in~\\cite{AP-geom}.\n\\end{proof}\n\n\n\n\\subsection{Tropical homology and cohomology groups} We recall the definition of tropical cohomology groups and refer to~\\cites{IKMZ, JSS, MZ, GS-sheaf} for more details.\n\nLet $X$ be an extended polyhedral space with a face structure. We start by recalling the definition of the multi-tangent and multi-cotangent spaces $\\SF_p(\\delta)$ and $\\SF^p(\\delta)$, respectively, for the face $\\delta$ of $X$. This leads to the definition of a chain, resp. cochain, complex which calculates the tropical homology, resp. cohomology, groups of $X$.\n\nFor a face $\\delta$ of $X$ and for any non-negative integer $p$, the \\emph{$p$-th multi-tangent} and the \\emph{$p$-th multicotangent space} of $X$ at $\\delta$ denoted by $\\SF_p(\\delta)$ and $\\SF^p(\\delta)$ are defined by\n\\[\\SF_{p}(\\delta)=  \\!\\!\\sum_{\\eta \\succ \\delta \\\\ \\sed(\\eta) = \\sed(\\delta) }\\!\\! \\bigwedge^p \\TT\\eta, \\qquad \\textrm{and} \\quad \\SF^{p}(\\delta) = \\SF_p(\\delta)^\\dual. \\]\n\nFor an inclusion of faces $\\gamma \\prec \\delta$ in $X$, we get maps $\\i_{\\delta\\succ\\gamma} \\colon \\SF_p(\\delta) \\to \\SF_p(\\gamma)$ and $\\i^*_{\\gamma\\prec\\delta} \\colon \\SF^p(\\gamma) \\to \\SF^p(\\delta)$.\n\n\\medskip\n\nFor a pair of non-negative integers $p,q$, define\n\\[C_{p,q}(X) := \\bigoplus_{\\delta \\in X \\\\ \\dims{\\delta} =q} \\SF_p(\\delta) \\]\nand consider the cellular chain complex, defined by using maps $\\i_{\\delta\\succ\\gamma}$ as above,\n\\[C_{p,\\bul}\\colon \\quad \\dots\\longrightarrow C_{p, q+1}(X) \\xrightarrow{\\partial^\\trop_{q+1}} C_{p,q}(X)  \\xrightarrow{\\ \\partial^\\trop_{q}\\ } C_{p,q-1} (X)\\longrightarrow\\cdots\\]\nThe tropical homology of $X$ is defined by\n\\[H_{p,q}^\\trop(X) := H_q(C_{p,\\bul}).\\]\nSimilarly, we have a cochain complex\n\\[C^{p,\\bul}\\colon \\quad \\dots\\longrightarrow C^{p, q-1}(X) \\xrightarrow{\\partial_\\trop^{q-1}} C^{p,q}(X)  \\xrightarrow{\\ \\partial_\\trop^{q}\\ } C^{p,q+1}(X) \\longrightarrow\\cdots\\]\nwhere\n\\[C^{p,q}(X) := \\bigoplus_{\\delta\\in X \\\\\n\\dims{\\delta}=q} \\SF^p(\\delta).\\]\nThe tropical cohomology of $X$ is defined by\n\\[H^{p,q}_\\trop(X) := H^q(C^{p,\\bul}).\\]\n\nIn the case $X$ comes with a rational structure, one can define tropical homology and cohomology groups with integer coefficients. In this case, for each face $\\delta$, the tangent space $\\TT\\delta$ has a lattice $N_\\delta$. One defines then\n\\[\\SF_{p}(\\delta,\\mathbb{Z})=  \\!\\!\\sum_{\\eta \\succ \\delta \\\\ \\sed(\\eta) = \\sed(\\delta) }\\!\\! \\bigwedge^p N_\\eta, \\quad \\textrm{and} \\qquad \\SF^{p}(\\delta, \\mathbb{Z}) = \\SF_p(\\delta,\\mathbb{Z})^{\\vee}.\\]\nSimilarly, we get complexes with $\\mathbb{Z}$-coefficients $C_{p,\\bul}^\\mathbb{Z}$ and $C^{p, \\bul}_\\mathbb{Z}$, and define\n\\[H_{p,q}^\\trop(X, \\mathbb{Z}) := H_q(C_{p,\\bul}^{\\mathbb{Z}}) \\qquad H^{p,q}_\\trop(X, \\mathbb{Z}) = H^q(C^{p,\\bul}_\\mathbb{Z}).\\]\n\nSimilarly, working with the cochain $C^{p, \\bullet}_c$ with \\emph{compact support}, one can define tropical cohomology groups with compact support. We denote them by $H^{p,q}_{\\trop,c}(X, A)$ with coefficients $A =\\mathbb{Z}, \\mathbb{Q},$ or $\\mathbb{R}$. If the coefficient field is not specified, it means we are working with real coefficients.\n\n\n\n\\subsection{Dolbeault cohomology and comparison theorem}\nIn this section we briefly recall the formalism of superforms and their corresponding Dolbeault cohomology on extended polyhedral complexes. The main references here are~\\cites{JSS, JRS19}, see also~\\cites{Lagerberg, CLD, GK17}.\n\nLet $V \\simeq \\mathbb{R}^n$ be a real vector space and denote by $V^\\dual$ its dual. We denote by $\\A^{p}$ the sheaf of differential forms of degree $p$ on $V$. Note that the sheaf $\\A^p$ is a module over the sheaf of rings of smooth functions $\\CC^\\infty$. The \\emph{sheaf of (super)forms of bidegree} $(p,q)$ denoted by $\\A^{p,q}$ is defined as the tensor product $\\A^p\\otimes_{\\CC^{\\infty}} \\A^q$. Choosing a basis $\\partial_{x_1}, \\dots, \\partial_{x_n}$ of $V$, we choose two copies of the dual vector space $V^\\dual$ with the dual basis $\\d' x_1, \\dots, \\d' x_n$ for the first copy and $\\d'' x_1, \\dots, \\d'' x_n$ for the second. In the tensor product $\\A^p\\otimes \\A^q$ the first basis is used to describe the forms in $\\A^p$ and the second basis for the forms in $\\A^q$. So for an open set $U \\subseteq V$, a section $\\alpha \\in \\A^{p,q}$ is uniquely described in the form\n\\[\\alpha = \\sum_{I, J \\subseteq [n]\\\\ \\card I = p , \\card J = q} \\alpha_{_{I, J}}\\,  \\d' x_I \\wedge \\d'' x_J\\]\nfor smooth functions $\\alpha_{I, J} \\in \\CC^\\infty(U)$. Here for a subset $I =\\{i_1, \\dots, i_\\ell\\} \\subseteq [n]$ with $i_1< i_2<\\dots<i_\\ell$, we denote by $\\d' x_I$ and $\\d'' x_I$ the elements of the two copies of $\\bigwedge^{\\ell} V^\\dual$ defined by\n\\[ \\d' x_I = \\d' x_{i_1} \\wedge \\dots \\wedge \\d' x_{i_\\ell} \\qquad \\textrm{and} \\qquad \\d'' x_I = \\d'' x_{i_1} \\wedge \\dots \\wedge \\d'' x_{i_\\ell},\\]\nrespectively. Moreover, for subsets $I, J \\subseteq [n]$, $\\d' x_I \\wedge \\d'' x_J$ is the element of $\\bigwedge^p V^\\dual \\otimes \\bigwedge^q V^\\dual$ given by $\\d' x_I \\otimes \\d'' x_J$. The usual rules of the wedge product apply. In particular, for any $i,j\\in[n]$,\n\\[ \\d'' x_j \\wedge \\d' x_i=-\\d' x_i \\wedge \\d'' x_j. \\]\n\n\\medskip\n\nObserving that $\\A^{p,q} \\simeq \\A^{p} \\otimes_\\mathbb{R} \\bigwedge^q V^\\dual$ and $\\A^{p,q} \\simeq \\bigwedge^p V^\\dual \\otimes \\A^q$, the collection of bigraded sheaves $\\A^{p,q}$ come with differential operators $\\d'\\colon \\A^{p,q} \\to \\A^{p+1,q}$ and $\\d'' \\colon \\A^{p,q} \\to \\A^{p,q+1}$ defined by $\\d' = \\d_p \\otimes \\id$ and $\\d'' = \\id\\otimes \\d_q$ where for non-negative integer $\\ell$, $\\d_\\ell \\colon \\A^\\ell \\to \\A^{\\ell+1}$ is the differentiation of smooth $\\ell$-forms. More explicitly, the differential map $\\d''$ is defined by its restriction to open sets $U \\subseteq V$\n\\[\\A^{p,q}(U) \\to \\A^{p,q+1}(U)\\]\nand sends a $(p,q)$-form $\\alpha = \\sum_{_{I, J}} \\alpha_{_{I, J}}\\,\\d' x_I \\wedge \\d'' x_J$ to\n\\[\\d'' \\alpha = \\sum_{I, J} \\sum_{\\ell=1}^n \\partial_{x_\\ell} \\alpha_{_{I,J}} \\,\\,\\d'' x_\\ell \\wedge \\d' x_I \\wedge \\d'' x_J = (-1)^p \\sum_{I, J} \\sum_{\\ell=1}^n \\partial_{x_\\ell} \\alpha_{_{I,J}} \\,\\,  \\d' x_I \\wedge (\\d'' x_\\ell\\wedge \\d'' x_J).  \\]\nThe operator $\\d'$ has a similar expression.\n\n\\medskip\n\nConsider now a fan $\\Sigma$ in $V$, and let $\\TP_\\Sigma$ be the corresponding partial compactification of $V$. We define the  \\emph{sheaf of bigraded $(p,q)$-forms} $\\A^{p,q}$ on $\\TP_\\Sigma$ as follows. For each pair of cones $\\tau\\prec\\sigma$ of $\\Sigma$, we have a natural projection $\\pi_{\\tau\\prec\\sigma}\\colon V^\\tau_\\infty \\to V^\\sigma_\\infty$. This projection induces a map $\\pi_{\\sigma\\succ\\tau}^*\\colon \\A^{p,q}(V^\\sigma_\\infty)\\to\\A^{p,q}(V^\\tau_\\infty)$. Let $U \\subseteq \\TP_\\Sigma$ be an open set. A section $\\alpha \\in \\A^{p,q}(U)$ is the data of the collection of $(p,q)$-forms $\\alpha^\\sigma \\in \\A^{p,q}(U \\cap V^{\\sigma}_\\infty)$, where $U \\cap V^\\sigma_\\infty$ is viewed as an open subset of the real vector space $V^\\sigma_\\infty \\simeq \\rquot{V}{\\TT\\sigma}$, so that the following compatibility at infinity is verified among these sections. For each pair of cones $\\tau\\prec\\sigma$ in $\\Sigma$, there exists a neighborhood $U'$ of $U\\cap V^\\sigma_\\infty$ in $U$ such that the restriction of $\\alpha^\\tau$ to $U'$ is given by the pullback ${\\pi^*_{\\sigma\\succ\\tau}} \\bigl(\\alpha^{\\sigma}\\bigr)\\rest{U'}$.\n\nThe differential operators $\\d'$ and $\\d''$ naturally extend to $\\TP_\\Sigma$ and define operators $\\d'\\colon \\A^{p, q}\\to \\A^{p+1, q}$ and $\\d''\\colon \\A^{p, q} \\to \\A^{p, q+1}$ for any pair of non-negative integers $(p, q)$.\n\n\\medskip\n\nLet now $X$ be a closed polyhedral complex in $\\TP_\\Sigma$, and denote $\\iota\\colon X  \\hookrightarrow \\TP_\\Sigma$ the inclusion. The sheaf of $(p,q)$-forms $\\A^{p,q}_X$ is defined as a quotient of the restriction $\\iota^* \\A^{p,q}_{\\TP_\\Sigma}$ of the sheaf of $(p,q)$-forms on $\\TP_\\Sigma$ to $X$ modulo those sections which vanish on the tangent space of $X$. Thus for an open set $U$ in $X$, a section $\\alpha \\in \\A^{p,q}_X(U)$ is locally around each point $x$ given by a section $\\alpha_x \\in \\A^{p,q}(U_x)$ for an open subset $U_x \\subseteq \\TP_\\Sigma$ with $U_x \\cap X  \\subseteq U$, and two sections $\\alpha$ and $\\beta$ in $\\A_X^{p,q}(X)$ are equivalent if for any point $x$ living in a face $\\sigma$ of $X$, the difference $\\alpha(x) - \\beta(x)$ vanishes on any $(p,q)$-multivector in $\\bigwedge^p \\TT \\sigma \\otimes \\bigwedge^q \\TT \\sigma$.\n\nThe definition above can be extended to any extended polyhedral complex.\n\nThe two differential operators $\\d'$ and $\\d''$ naturally extend to $\\TP_{X_\\sminfty}$ and we get $\\d'\\colon \\A_X^{p, q}\\to \\A_X^{p+1, q}$ and $\\d''\\colon \\A_X^{p, q} \\to \\A_X^{p, q+1}$ for any pair of non-negative integers $(p, q)$.\n\n\\medskip\n\nThe above definition only depends on the support of $X$ and not on the chosen polyhedral structure on this support. As a consequence, if $\\mathfrak X$ is any polyhedral space, the sheaf $\\A^{p,q}_\\mathfrak X$ is well-defined.\n\n\\medskip\n\nThe \\emph{Dolbeault cohomology group $H^{p,q}_{\\mathrm{Dol}}(X)$ of bidegree $(p,q)$} is by definition the $q$-th cohomology group of the cochain complex\n\\[\\bigl(\\A_X^{p,\\bullet}(X), \\d''\\bigr)\\colon \\qquad \\A_X^{p,0}(X) \\to \\A^{p,1}_X(X) \\to \\dots \\to \\A^{p,q-1}_X(X) \\to \\A^{p,q}_X(X) \\to \\A^{p,q+1}_X(X) \\to \\cdots \\]\n\nSimilarly, we define Dolbeault cohomology groups with compact support $H^{p,q}_{\\mathrm{Dol},c}(X)$ as the $q$-th cohomology group of the cochain complex\n\\[\\bigl(\\A_{X,c}^{p,\\bullet}(X), \\d''\\bigr)\\colon \\qquad \\A_{X,c}^{p,0}(X) \\to \\A^{p,1}_{X,c}(X) \\to \\cdots \\]\nwhere $\\A^{p,q}_{X, c}(X)$ is the space of $(p,q)$-forms whose support forms a compact subset of $X$.\n\nDenote by $\\Omega_X^p$ the kernel of the map $\\d''\\colon \\A_X^{p,0} \\to \\A^{p,1}_X$. We call this the sheaf of \\emph{holomorphic $p$-forms} on $X$.\nBy Poincar\\'e lemma for superforms, proved in~\\cites{Jell, JSS}, the complex $(\\A^{p,\\bullet}, \\d'')$ provides an acyclic resolution of the sheaf $\\Omega_X^p$. We thus get the isomorphism\n\\[H^{p,q}_\\mathrm{Dol}(X) \\simeq H^q(X, \\Omega_X^p),\\]\nand similarly,\n\\[H^{p,q}_{\\mathrm{Dol}, c}(X) \\simeq H^q_c(X, \\Omega_X^p).\\]\nMoreover, we have the following comparison theorem from~\\cite{JSS}.\n\\begin{thm}[Comparison theorem]\\label{thm:comparison}\nFor any extended polyhedral complex $X$, we have\n\\[H^{p,q}_\\trop(X) \\simeq H^{p,q}_\\mathrm{Dol}(X) , \\qquad \\textrm{and} \\qquad H^{p,q}_{\\trop,c}(X) \\simeq H^{p,q}_{\\mathrm{Dol},c}(X).\\]\n\\end{thm}\n\n\n\\bigskip\n\n\n\n\\subsection{Matroids} \\label{sec:matroid} We first recall basic definitions involving matroids, and refer to relevant part of \\cite{Oxl11} for more details.\n\n\\medskip\n\nFor a set $E$, a subset $A \\subseteq E$ and an element $a\\in E$, if $a\\in A$, we denote by $A-a$ the set $A \\setminus \\{a\\}$; if $a\\notin A$, we denote by $A+a$ the set $A \\cup\\{a\\}$.\n\n\\medskip\n\nA matroid can be defined in different \\emph{equivalent} manners, for example by specifying its collection of \\emph{independent sets}, or its collection of \\emph{bases}, or its collection of \\emph{flats}, or still by giving its \\emph{rank function}. In each case, the data of one of these determines all the others. The definition of a matroid with respect to independent sets is as follows.\n\n\\begin{defi}[Matroid: definition with respect to independent sets] \\label{defi:matroid}\nA \\emph{matroid} $\\Ma$ is a pair consisting of a finite set $E$ called the \\emph{ground set of $\\Ma$} and a family $\\mathfrak I$ of subsets of $E$ called the \\emph{family of independent sets of $\\Ma$} which verifies the following three axiomatic properties:\n\\begin{enumerate}\n\\item $\\emptyset\\in\\mathfrak I$.\n\\item (hereditary property) $\\mathfrak I$ is stable under inclusion, in the sense that if $J\\subset I$ and $I\\in\\mathfrak I$, then $J\\in\\mathfrak I$.\n\\item (augmentation property) if $I, J\\in\\mathfrak I$ and if $\\card J<\\card I$, then there exists an element $i\\in I\\setminus J$ such that $J+i\\in\\mathfrak I$. \\label{defi:matroid:augmentation_property}  \\qedhere\n\\end{enumerate}\n\\end{defi}\nIf needed, we write $\\Ma=(E, \\mathfrak I)$ in order to emphasize that $\\Ma$ is defined on the ground set $E$ and has independent sets $\\mathfrak I \\subseteq 2^E$.\n\n\\begin{example}\\label{ex:representable} An example of a matroid is given by a collection of vectors $v_1, \\dots, v_m$ in a vector space $H \\simeq k^n$, over a field $k$. Such a collection defines a matroid $\\Ma$ on the ground set $[m]$ whose independent $\\mathfrak I$ are all the subsets $I \\subseteq [m]$ such that the corresponding vectors $v_i$ for $i\\in I$ are linearly independent. A matroid $\\Ma$ which is defined by a collection of vectors in a vector space is said to be \\emph{representable}. By an interesting recent result of Peter Nelson~\\cite{Nelson}, we know that \\emph{almost any matroid is non representable over any field}. More precisely, if one denotes by $\\epsilon_m$ the number of representable matroids on the ground set $[m]$ divided by the total number of matroids on the same ground set, then the numbers $\\epsilon_m$ tend to zero when $m$ goes to infinity; Nelson gives estimates for all these values.\n\\end{example}\n\nFor a matroid $\\Ma=(E,\\mathfrak I)$, the \\emph{rank function} $\\mathrm{rk}\\colon 2^E \\to \\mathbb N \\cup\\{0\\}$ is defined as follows. For a subset $A \\subseteq E$, the \\emph{rank of $A$} is defined by\n\\[ \\mathrm{rk}(A):=\\max_{\\substack{I \\in \\mathfrak I \\textrm{ with }I \\subseteq A}} \\, \\card I. \\]\nWe define the \\emph{rank of $\\Ma$} to be the integer number $\\mathrm{rk}(E)$. The rank function satisfies what is called the \\emph{submodularity property}, meaning\n\\[\\forall A, A' \\subseteq E, \\qquad \\mathrm{rk}(A) + \\mathrm{rk}(A') \\leq \\mathrm{rk}(A\\cap A') + \\mathrm{rk}(A \\cup A').\\]\nThe data of $E$ and rank function $\\mathrm{rk}$ determines $\\mathfrak I$ as the set of all subsets $I \\subseteq E$ which verify $\\mathrm{rk}(I) = \\card{I}$.\n\nA maximal independent set in $\\Ma$ is called a \\emph{basis} of $\\Ma$. The collection of bases of $\\Ma$ is denoted by $\\mathfrak B(\\Ma)$.\n\nThe \\emph{closure} of a subset $A \\subseteq E$ is the set\n\\[ \\cl(A):=\\{a \\in E \\textrm{ such that }  \\mathrm{rk}(A + a)=\\mathrm{rk}(A)\\}. \\]\nA subset $F \\subseteq E$ is called a \\emph{flat} of $\\Ma$ if $\\cl(F) = F$. Flats are also equally called \\emph{closed set} and the collection of flats of the matroid $\\Ma$ is denoted by $\\Cl(\\Ma)$.\n\n\\begin{example}[Example~\\ref{ex:representable} continued] For a matroid $\\Ma$ representable by a collection of vectors $v_1, \\dots, v_m$ in a vector space $ H \\simeq k^n$, the above notions coincide with the usual linear algebra notions. For example the rank $\\mathrm{rk}(A)$ is the dimension of the vector subspace generated by the set of vectors $v_i$ for $i\\in A$. A flat $F$ of $\\Ma$ is a subset of $[m]$ such that $\\{v_i\\}_{i\\in F}$ is the set of the vectors in the collection which live in a vector subspace of $H$.\n\\end{example}\n\nAn element $e\\in E$ is called a \\emph{loop} if $\\{e\\}$ is not an independent set, i.e., if $\\mathrm{rk}(\\{e\\})=0$. Two elements $e$ and $e'$ of $E$ are called \\emph{parallel} if $\\mathrm{rk}(\\{e, e'\\}) = \\mathrm{rk}(\\{e\\}) = \\mathrm{rk}(\\{e'\\}) = 1$. A matroid is called \\emph{simple} if it neither contains any loop nor any parallel elements.\n\nAn element $e\\in E$ is called a \\emph{coloop} if $\\mathrm{rk}(E-e)=\\mathrm{rk}(E)-1$, equivalently, if $E-e$ is a flat. A \\emph{proper element} of $\\Ma$ is an element which is neither a loop nor a coloop.\n\n\n\n\\subsection{Bergman fans} For a simple matroid $\\Ma$ on a ground set $E$, the Bergman fan of $\\Ma$ denoted by $\\Sigma_\\Ma$ is defined as follows.\n\nFirst, we fix some notations. We denote by $\\{\\e_i\\}_{i\\in E}$ the standard basis of $V$. For a subset $A \\subseteq E$, we denote by $\\e_A$ the sum $\\sum_{i\\in A} \\e_i$ in $V$.\n\nConsider the real vector space $V := \\rquot{\\mathbb{R}^E}{\\mathbb{R} \\e_E}$. It has a full rank lattice $N = \\rquot{\\mathbb{Z}^E}{\\mathbb{Z} \\e_E}$. By an abuse of the notation, we denote by $\\e_A$ the projection of $\\e_A=\\sum_{i\\in A}\\e_i$ in $V$. In particular, we have $\\e_E =0$.\n\nThe \\emph{Bergman fan of $\\Ma$} denoted by $\\Sigma_\\Ma$ is a rational fan in $N_\\mathbb{R} =V$ of dimension $\\mathrm{rk}(\\Ma) -1$ defined as follows. A \\emph{flag of proper flats $\\mathscr{F}$ of $\\Ma$ of length $\\ell$} is a collection\n\\[\\mathscr{F}\\colon \\quad \\emptyset \\neq F_1 \\subsetneq F_2 \\subsetneq \\dots \\subsetneq F_{\\ell} \\neq E\\]\nconsisting of flats $F_1, \\dots, F_\\ell$ of $\\Ma$.\n\nFor each such flag $\\mathscr{F}$, we define the rational cone $\\sigma_\\mathscr{F} \\subseteq V$ as the cone generated by the vectors $\\e_{F_1}, \\e_{F_2}, \\dots, \\e_{F_\\ell}$, so\n\\[\\sigma_\\mathscr{F} := \\Bigl\\{ \\lambda_1 \\e_{F_1} + \\dots + \\lambda_\\ell \\e_{F_\\ell} \\Bigst \\lambda_1, \\dots, \\lambda_\\ell \\geq0\\Bigr\\}.\\]\nNote that the dimension of $\\sigma_\\mathscr{F}$ is equal to the length of $\\mathscr{F}$.\n\nThe Bergman fan of $\\Ma$ is the fan consisting of all the cones $\\sigma_\\mathscr{F}$ for $\\mathscr{F}$ a flag of proper lattices of $\\Ma$:\n\\[\\Sigma_\\Ma := \\Bigl\\{\\, \\sigma_\\mathscr{F} \\Bigst \\mathscr{F} \\textrm{ flag of proper flats of }\\Ma\\,\\Bigr\\}.\\]\nFrom the definition, it follows easily that $\\Sigma_\\Ma$ has pure dimension $\\mathrm{rk}(\\Ma)-1$.\n\n\\medskip\n\nA subset $S$ in a real vector space $W$ is called a \\emph{Bergman support} if there exists a matroid $\\Ma$ on a ground set $E$ and a linear map $\\phi\\colon W \\to \\rquot{\\mathbb{R}^E}{\\mathbb{R} \\e_E}$ which induces a bijection $\\phi\\rest S\\colon S \\to \\supp{\\Sigma_\\Ma}$. A fan $\\Sigma$\nin a real vector space is called a \\emph{Bergman fan} if the support $\\supp{\\Sigma}$ of $\\Sigma$ is a Bergman support.\n\nA \\emph{rational Bergman fan} is a fan in a vector space $W$ which is rational with respect to a full rank lattice $N_W$ in $W$ such that there exists a $\\mathbb{Z}$-linear map $\\phi\\colon N_W \\to \\rquot{\\mathbb{Z}^E}{\\mathbb{Z}\\e_E}$ which, once extended to a linear map $W \\to \\rquot{\\mathbb{R}^E}{\\mathbb{R} \\e_E}$, induces an isomorphism\n$\\supp{\\Sigma} \\to \\supp{\\Sigma_\\Ma}$.\n\n\\begin{example} Any complete fan $\\Sigma$ in a real vector space $\\mathbb{R}^n$ is a Bergman fan. In fact, denoting by $U_{n+1}$ the \\emph{uniform matroid on the ground set $E=[n+1]$} whose collection of independent sets is $\\mathfrak I = 2^{[n+1]}$, we see that $\\mathbb{R}^n \\simeq \\rquot{\\mathbb{R}^{n+1}}{\\mathbb{R} \\e_{E}}$ is a Bergman support. Therefore, $\\Sigma$ is a Bergman fan. Moreover, $\\Sigma$ is a rational Bergman fan if it is rational (in the usual sense) with respect to the lattice $\\mathbb{Z}^n$.\n\\end{example}\n\n\\begin{example}[Example~\\ref{ex:representable} continued] Let $\\Ma$ be a matroid on the ground set $[m]$ representable by a collection of non-zero vectors $v_1, \\dots, v_m$ in a vector space $V$ over $k$. Viewing $v_1, \\dots, v_m$ as elements of the dual vector space of $V^\\dual$, we get a collection of hyperplanes $H_1, \\dots, H_m \\subset V^\\dual$ with $H_j = \\ker\\Bigl( v_j \\colon  V^\\dual \\to k\\Bigr)$ for each $j$, which leads to a hyperplane arrangements denoted, by abusing of the notation, by $H_1, \\dots, H_m \\subseteq \\P(V^\\dual)$ in the projective space $\\P(V^\\dual)$.\n\nDenote by $X$ the complement in $\\P(V^\\dual)$ of the union $H_1 \\cup \\dots \\cup H_m$. The collection of linear forms $v_j$ gives an embedding of $X\\hookrightarrow \\rquot{\\G_m^{m}}{\\G_m}$, where $\\G_m$ is the algebraic torus over $k$ whose set of $K$-points for each field extension $K$ of $k$ is $K^\\times$, and the quotient $\\rquot{\\G_m^{m}}{\\G_m}$ means the quotient of $\\G_m^{m}$ by the diagonal action of $\\G_m$. The tropicalization of $X$ in the torus $\\rquot{\\G_m^{m}}{\\G_m}$ coincides with the Bergman fan $\\Sigma_\\Ma$ for the representable matroid $\\Ma$ which is associated to the collection of vectors $v_1, \\dots, v_m$. See~\\cites{AK, MS} for more details.\n\\end{example}\n\n\\medskip\n\nThe category of Bergman fans is closed under product as the following proposition shows.\n\n\\begin{prop}\\label{prop:productBergman} The product of any two Bergman supports is a Bergman support. In particular, the product of two Bergman fans is again Bergman.\n\nMore precisely, for two matroids $\\Ma$ and $\\Ma'$, we have\n\\[\\supp{\\Sigma_{\\Ma}} \\times \\supp{\\Sigma_{\\Ma'}} = \\supp{\\Sigma_{\\Ma\\vee\\Ma'}}\\]\nwhere $\\Ma\\vee \\Ma'$ is a parallel connection of $\\Ma$ and $\\Ma'$.\n\\end{prop}\n\nWe recall the definition of \\emph{parallel connection} between two matroids, \\cf.~\\cite{Oxl11}*{Chapter 7}, and refer to~\\cite{AP} for the proof of this proposition.\n\nA \\emph{pointed matroid} is a matroid $\\Ma$ on a ground set $E$ with a choice of a distinguished element $\\ast = \\ast_\\Ma$ in $E$.\n\nFor two pointed matroids $\\Ma$ and $\\Ma'$ on ground sets $E$ and $E'$ with distinguished elements $\\ast_\\Ma \\in E$ and $\\ast_{\\Ma'} \\in E'$, respectively, the \\emph{parallel connection} of $\\Ma$ and $\\Ma'$, that we can also call the \\emph{wedge sum} of $\\Ma$ and $\\Ma'$, denoted by $\\Ma \\vee \\Ma'$, is by definition the pointed matroid on the ground set $E \\vee E' = \\rquot{E \\sqcup E'}{(\\ast_\\Ma=\\ast_{\\Ma'})}$, the wedge sum of the two pointed sets $E$ and $E'$, with distinguished element $\\ast = \\ast_\\Ma = \\ast_{\\Ma'}$, and with the following collection of bases:\n\n\\begin{align*}\n\\mathfrak B( \\Ma \\vee \\Ma') =& \\Bigl\\{ \\,B \\cup B' \\st B \\in \\mathfrak B(\\Ma), B' \\in \\mathfrak B(\\Ma') \\,\\textrm{ with }\\,\\ast_\\Ma\\in B\\, \\textrm{ and }\\,  \\ast_{\\Ma'}\\in B'\\Bigr \\} \\\\\n& \\cup  \\Bigl\\{ \\,B \\cup B' \\setminus\\{\\ast_\\Ma\\} \\st B \\in \\mathfrak B (\\Ma), B' \\in \\mathfrak B(\\Ma') \\, \\textrm{ with }\\, \\ast_\\Ma\\in B \\, \\textrm{ and }\\, \\ast_{\\Ma'}\\notin B'\\Bigr \\}\\\\\n& \\cup  \\Bigl\\{ \\,B \\cup B' \\setminus\\{\\ast_{\\Ma'}\\}\\st B \\in \\mathfrak B(\\Ma), B' \\in \\mathfrak B(\\Ma')\\, \\textrm{ with }\\, \\ast_\\Ma \\notin B\\, \\textrm{ and }\\,  \\ast_{\\Ma'}\\in B'\\Bigr \\}.\n\\end{align*}\n\nA parallel connection of two matroids $\\Ma$ and $\\Ma'$ is the wedge sum of the form $(\\Ma, e) \\vee (\\Ma', e')$ for the choice of two elements $e\\in \\Ma$ and $e'\\in \\Ma'$, which make pointed matroids out of them.\n\n\n\n\\subsection{Tropical cycles and divisors} Let $V =N_\\mathbb{R} \\simeq \\mathbb{R}^n$ be a real vector space of dimension $n$ with a full rank lattice $N$. Let $Y$ be a rational polyhedral complex of pure dimension $d$ in $V$. For each facet $\\sigma$ of $Y$, suppose we are given a \\emph{weight} which is an integer $w(\\sigma)$. Let $C := (Y, w)$ be the weighted polyhedral complex $Y$ with the weight function on its facets $w$. The weighted polyhedral complex $C$ is called a \\emph{tropical cycle} if the following \\emph{balancing condition} is verified:\n\\[\\forall \\tau\\in Y\\text{ of dimension $d-1$},\\qquad  \\sum_{\\sigma \\ssupface \\tau}  w(\\sigma) \\e_{\\sigma\/\\tau}  = 0 \\in N^{\\tau}.\\]\nHere $\\e_{\\sigma\/\\tau}$ is the primitive vector of the ray $\\rho_{\\sigma\/\\tau}$ corresponding to $\\sigma$ in $N^\\tau_\\mathbb{R}$. The following is a direct consequence of the the definition of the Bergman fan given in the previous section.\n\n\\begin{prop} The Bergman fan $\\Sigma_\\Ma$ of any matroid with weights equal to one on facets is a tropical cycle. More generally any Bergman fan is a tropical cycle.\n\\end{prop}\n\nA \\emph{tropical regular function} $f$ on a tropical cycle $C$ is a piecewise affine function with integral slopes which is the restriction to $C$ of a concave function on $\\mathbb{R}^n$. In other words, it is a function of the form\n\\[f = \\min_{i \\in I} \\,\\bigl\\{ \\langle \\alpha_i, \\,\\cdot\\, \\rangle + \\beta_i\\bigr\\}\\]\nfor a finite index set $I$ and $\\alpha_i \\in \\mathbb{Z}^n$, $\\beta_i\\in \\mathbb{Q}$.\n\nAny tropical regular function induces a subdivision of the tropical cycle $C$. On each face $\\delta$ of $C$, the new polyhedral structure consists of set of the form $\\{x\\in\\delta \\st f(x) = g(x) \\}$ where $g$ is an affine linear function such that $g(x)\\geq f(x)$ for every $x$. Keeping the same weight on facets, this leads to a new tropical cycle that we denote by $C_1$ and note that $C$ and $C_1$ have the same support and same weight function.\n\nGiven a tropical regular function $f$ on $C$, we can consider the graph $\\Gamma_f$ of $f$ in $V \\times \\mathbb{R} \\simeq \\mathbb{R}^{n+1},$ with the polyhedral structure obtained by that of the subdivided $C_1$. The graph $\\Gamma_f$ is not necessary a tropical cycle in $V\\times \\mathbb{R}$ anymore, as it might have unbalanced codimension one faces. In this case, we can add new faces to $\\Gamma_f$ to obtain a tropical cycle. For this, consider a codimension one face $\\delta$ of $C_1$ which turns to be unbalanced in the graph $\\Gamma_f$. Consider the  polyhedron $E_\\delta$ defined by\n\\[E_\\delta := \\Bigl\\{ (x, t) \\st x \\in \\delta \\text{ and } t \\geq f(x)\\Bigr\\} \\subset V \\times \\mathbb{R}.\\]\nDefine $\\widetilde C$ as the union of $\\Gamma_f$ with the polyhedra $E_\\delta$ and there faces, one polyhedron for each unbalanced codimension one face $\\delta$ of $\\Gamma_f$.  Note that for each such face $\\delta$, the primitive vector $\\e_{E_\\delta\/\\delta}$ is the one given by the last vector $\\e_{n+1}$ in the standard basis of $V\\times \\mathbb{R} \\simeq \\mathbb{R}^{n+1}$.\n\nOne verifies there is a unique integer weight $w(E_\\delta)$ which we can associate to new faces $E_\\delta$ in order to make $\\widetilde C$ a tropical cycle in $V\\times \\mathbb{R}$.  The new tropical cycle in $V\\times \\mathbb{R}$ is called tropical modification of $C$ along $\\div_C(f)$, where $\\div_C(f)$ is the union of the $\\delta$ in $C_1$ of codimension one which happens to be unbalanced when seen in $\\Gamma_f$.\n\n\n\n\\subsection{Tropical modifications of Bergman fans} \\label{sec:modification}\n\nFor a matroid $\\Ma$ on a ground set $E$ and for an element $a \\in E$, we denote by $\\Ma \\del a$ the matroid obtained by the \\emph{deletion of $a$}. This is the matroid on ground set $E - a$ whose independent sets are those of $\\Ma$ which do not contain $a$. The rank function of $\\Ma\\del a$ is thus that of $\\Ma$ restricted to the set $E-a$. The flats of $\\Ma\\del a$ are all the sets $A = F\\setminus\\{a\\}$ for a flat $F$ of $\\Ma$, so\n\\[\\Cl(\\Ma \\del a) := \\Bigl\\{F \\subseteq E - a \\st \\textrm{either $F$ or $F+a$ is a flat of $\\Ma$} \\Bigr\\}.\\]\n\nFor an element $a \\in E$, the matroid obtained by \\emph{contraction of $a$ in $\\Ma$} is the matroid denoted by $\\Ma\\contr a$ on the ground set $E-a$ whose independent sets are precisely those subsets $I \\subseteq E -a$ such that $I+a$ is in independent set in $\\Ma$.\nIt follows that the flats of $\\Ma\\contr a$ are obtained by removing $a$ from the flats of $\\Ma$ which contain $a$, i.e.,\n\\[\\Cl(\\Ma \\contr a) := \\Bigl\\{F \\subseteq E - a \\st \\textrm{$F+a$ is a flat of $\\Ma$} \\Bigr\\}.\\]\n\nLet $N=\\rquot{\\mathbb{Z}^E}{\\mathbb{Z} \\e_E}$, $M = N^\\vee$ the dual lattice, and let $N^a := \\rquot{\\mathbb{Z}^{E-a}}{\\mathbb{Z} \\e_{_{E-a}}} = \\rquot{N}{\\mathbb{Z} \\e_a}$ and $M^a = N^{a\\vee}$.\nThe two Bergman fans $\\Sigma_{\\Ma \\del a}$ and $\\Sigma_{\\Ma \\contr a}$ both live in the real vector space $N^a_\\mathbb{R}.$ Moreover, from the description of the flats of these matroids, it is clear that $\\supp{\\Sigma_{\\Ma \\contr a}} \\subseteq \\supp{\\Sigma_{\\Ma \\del a}}$. In addition, if $a$ is proper, then we have $\\mathrm{rk}(\\Ma \\del a) = \\mathrm{rk}(\\Ma \\contr a)-1$. In this case, $\\supp{\\Sigma_{\\Ma \\contr a}}$ is of codimension one in $\\supp{\\Sigma_{\\Ma \\del a}}$.\n\n\\begin{prop} Assume $\\Ma$ is a simple matroid and $a\\in\\Ma$ is a proper element. Let $\\Delta'$ be a Bergman fan with support $ \\supp{\\Delta'} = \\supp{\\Sigma_{\\Ma \\contr a}} $ and let $\\Sigma'$ be a Bergman fan with support $\\supp{\\Sigma'} = \\supp{\\Sigma_{\\Ma \\del a}}$ such that $\\Delta' \\subseteq \\Sigma'$. Denote by $\\Sigma$ the tropical modification of $\\Sigma'$ along the cycle $\\Delta'$. Then $\\Sigma$ is a Bergman fan with support $\\supp{\\Sigma} = \\supp{\\Sigma_\\Ma}$. Moreover, if $\\Delta'$ and $\\Sigma'$ are unimodular, then so is $\\Sigma'$.\n\\end{prop}\n\\begin{proof}\nThis follows from the observation that $\\Sigma_{\\Ma}$ is a subdivision the tropical modification of $\\Sigma_{\\Ma\\del a}$ along $\\Sigma_{\\Ma \\contr a}$. More detail is given in \\cite{AP-geom}.\n\\end{proof}\n\n\\medskip\n\n\n\n\\subsection{Tropical smoothness of the support} An extended polyhedral complex $X$ is said to have a \\emph{tropically smooth,} or simply \\emph{smooth, support} if its support $\\supp X$ is locally isomorphic to a product $\\eR^a \\times \\supp \\Sigma$ for a non-negative integer $a$ and a Bergman fan $\\Sigma$.\n\n\\medskip\n\nWe prove in~\\cite{AP-geom} that the canonical compactification of any unimodular Bergman fan has smooth support. In fact, we have the following more general result.\n\n\\begin{thm}\\label{thm:smoothness-compact} Let $X$ be a rational polyhedral complex in $N_\\mathbb{R}$ with smooth support. Suppose that the recession fan $X_\\infty$ is unimodular. Then the canonical compactification $\\comp X$ of $X$ has smooth support. Moreover, if the polyhedral structure on $X$ is unimodular, then the induced extended polyhedral structure on $\\comp X$, given in Theorem~\\ref{thm:orbit-stratum-correspondence}, is unimodular as well.\n\\end{thm}\n\nThis theorem is proved in~\\cite{AP-geom}.\n\n\n\n\n\n\n\\section{Local Hodge-Riemann and Hard Lefschetz} \\label{sec:local}\nThe aim of this section is to study the local situation and to prove Theorem~\\ref{thm-intro:HL-HR-local}.\n\nThe results in this section are valid for integral, rational, or real Chow rings of simplicial Bergman fans when they are definable. Since our primary objective in this paper is to establish global versions of these results, and we do not have tried to control the torsion part of the tropical cohomology for more general tropical varieties, to simplify the presentation, we will only consider the case of rational fans and we suppose the coefficient ring $\\corps$ be either $\\mathbb{Q}$ or $\\mathbb{R}$.\n\n\n\n\\subsection{Basic definitions and results}\n\nIn this section, we review some basic results concerning Hodge theory of finite dimensional algebras.\n\n\\medskip\n\nLet $A^\\bul = \\bigoplus_{i\\in \\mathbb Z} A^i$ be a graded vector space of finite dimension over $\\corps$ such that the graded pieces of negative degree are all trivial.\n\n\\medskip\n\n\\begin{defi}[Poincar\\'e duality]  We say that $A^\\bul$ verifies\n\\begin{itemize}[label=-]\n\\item  \\emph{weak Poincar\\'e duality for an integer $d$} denoted by $\\WPD^d(A^\\bul)$ if for any integer $k$, we have the equality $\\dim_{\\corps} A^k = \\dim_{\\corps} A^{d-k}$.\n\\end{itemize}\nLet $\\Phi \\colon A^\\bul\\times A^\\bul \\to \\corps$ be a symmetric bilinear form on $A^\\bul$. We say that the pair $(A^\\bul, \\Phi)$ verifies\n\\begin{itemize}[label=-]\n\\item  \\emph{Poincar\\'e duality for an integer $d$} that we denote by $\\PD^d(A^\\bul, \\Phi)$ if $\\Phi$ is non-degenerate and for any integer $k$, $A^k$ is orthogonal to $A^l$ for all $l \\neq d-k$. \\qedhere\n\\end{itemize}\n\\end{defi}\n\n\\medskip\n\nLet $A^\\bul$ be a graded $\\corps$-vector space of finite dimension as above, let $d$ be a positive integer, and let $Q$ be a symmetric bilinear form on the truncation $\\bigoplus_{k\\leq \\frac d2} A^k$ such that $A^i$ and $A^j$ are orthogonal for distinct pair of non-negative integers $i,j \\leq \\frac d2$.\n\n\\begin{defi}[Hard Lefschetz] Notations as above, we say that the pair $(A^\\bul,Q)$ verifies\n\\begin{itemize}[label=-]\n\\item  \\emph{Hard Lefschetz property} denoted $\\HL^d(A^\\bul, Q)$ if $A^\\bul$ verifies $\\WPD^d(A^\\bul)$ and $Q$ is non-degenerate.\n\\end{itemize}\n\\end{defi}\n\nIn the sequel, for any $k\\leq d\/2$, we denote by $Q_k$ the restriction of $Q$ to $A^k$.\n\n\\begin{defi}[Hodge-Riemann] We say the pair $(A^\\bul, Q)$ verifies\n\\begin{itemize}[label=-]\n\\item \\emph{Hodge-Riemann relations} $\\HR^d(A^\\bul, Q)$ if $A^\\bul$ verifies $\\WPD^d(A^\\bul)$ and moreover, for each non-negative integer $k\\leq \\frac d2$, the signature of $Q_k$ is given by the sum\n\\[ \\sum_{i=0}^k (-1)^i\\bigl(\\dim(A^i)-\\dim(A^{i-1})\\bigr). \\qedhere \\]\n\\end{itemize}\n\\end{defi}\n\n\\begin{remark} As we will show in Proposition \\ref{prop:HR} below, the above definitions are in fact equivalent to the ones given in the introduction.\n\\end{remark}\n\n\n\\subsubsection{Bilinear form associated to a degree map} Assume that $A^\\bul$ is a unitary graded $\\corps$-algebra with unit $1_{A^\\bul}$ and suppose $A^0 = \\corps \\cdot 1_{A^\\bul}$. Moreover, suppose that we are given a linear form $\\deg\\colon A^d \\to \\corps$ called the \\emph{degree map} which we extend to the entire algebra $A^\\bul$ by declaring it to be zero on each graded piece $A^k$ for $k\\neq d$. In this way, we get a bilinear form on $A^\\bul$ denoted by $\\Phi_{\\deg}$ and defined by\n\\[ \\begin{array}{rccl}\n  \\Phi_{\\deg} \\colon & A^\\bul \\times A^\\bul & \\to & \\corps, \\\\\n  & (a, b) & \\mapsto & \\deg(ab).\n\\end{array} \\]\n\n\n\\subsubsection{Lefschetz operator and its associated bilinear form}\\label{sec:lefschetz} Notations as in the preceding paragraph, let now $L\\colon A^\\bul\\to A^{\\bul+1}$ be a morphism of degree one which sends any element $a$ of $A^\\bul$ to the element $La\\in A^{\\bul+1}$. Let $\\Phi$ be a symmetric bilinear form on $A^\\bul$. For example, if we have a degree map on $A^\\bul$, we can take $\\Phi = \\Phi_{\\deg}$.\n\n\\medskip\n\nWe define a bilinear form $Q_{L,\\Phi}$ on $A^\\bul$ by setting\n\\[\\forall\\,\\, a\\in A^k,\\, b\\in A^j, \\qquad Q_{L,\\Phi}(a, b):=\\Phi(a,L^{d-2k}b)\\]\nfor all pairs of integers $k,j\\leq d\/2$. We furthermore require $A^j$ and $A^k$ to be orthogonal for $j\\neq k$. In the case $\\Phi = \\Phi_{\\deg}$, we will have $Q_{L,\\Phi}(a, b) = \\deg(a\\cdot L^{d-2k}b)$ and this latter requirement will be automatic.\n\n\\medskip\n\nAssuming that $L$ is auto-adjoint with respect to $Q_{L, \\Phi}$, i.e., $\\Phi(a,L b) = \\Phi(La, b)$ for all $a,b \\in A^\\bul$, we get a symmetric bilinear form $Q = Q_{L, \\Phi}$ on $A^\\bul$.\n\n\\medskip\n\nIf $L$ is given by multiplication by an element $\\ell\\in A^1$ and $\\Phi=\\Phi_{\\deg}$, we write $\\ell$ instead of $L$ for the corresponding map, and write $Q_{\\ell}$ for the corresponding bilinear form, which is then automatically symmetric: in fact, we have\n\\[Q_{\\ell}(a,b)=\\deg(\\ell^{d-2k}ab)\\]\nfor any pair of elements $a,b \\in A^k$. In this case, we write $\\HL^d(A^\\bul, \\ell)$ and $\\HR^d(A^\\bul, \\ell)$ instead of $\\HL^d(A^\\bul, Q_{\\ell,\\Phi_{\\deg}})$ and $\\HR^d(A^\\bul, Q_{\\ell,\\Phi_{\\deg}})$, respectively. If there is no risk of confusion, we furthermore omit the mention of the top dimension $d$ in the above notations. Moreover, we sometimes use $\\deg$ instead of $\\Phi_{\\deg}$ in order to simplify the notation.\n\n\n\\subsubsection{Primitive parts induced by a Lefschetz operator} For a Lefschetz operator $L\\colon A^\\bul \\to A^{\\bul+1}$, as in the preceding section, and for a non-negative integer $k\\leq \\frac d2$,\nthe \\emph{primitive part of $A^k$ with respect to $L$} denoted by $A^k_{\\prim, L}$ is by definition the kernel of the map\n\\[L^{r-2k+1} \\colon A^{k} \\longrightarrow A^{k-r+1}.\\]\n\nIf $L$ is given by an element $\\ell \\in A^1$, we denoted by $A^k_{\\prim, \\ell}$ the primitive part of $A^k$ with respect to $\\ell$.\n\nIf there is no risk of confusion, and $L$ or $\\ell$ are understood from the context, we simply drop them and write $A^k_{\\prim}$ for the primitive part of $A^k$.\n\n\n\\subsubsection{Basic implications and consequences} For future use, we state here basic results around these notions and some of their consequences.\n\nWe start by the following proposition which gathers several useful implications and relations between the properties introduced above.\n\nWe use the notations of the preceding paragraph. We suppose the Lefschetz operator $L$ is auto-adjoint with respect to $\\Phi$ so that the corresponding bilinear form becomes symmetric.\n\n\\begin{prop}  \\label{prop:HR} Notations as above, the following properties are verified.\n\n\\let\\olditem\\item\n\\renewcommand{\\item}{\\stepcounter{equation}\\olditem}\n\\begin{enumerate}[label=\\textnormal{(\\theequation)}, ref={\\theequation}, labelindent={0pt}, leftmargin=*]\n\\item \\label{prop:HR:implication} \\ \\vspace*{-1.3em} \\[ \\HR^d(A^\\bul, L, \\Phi)\\Longrightarrow\\HL^d(A^\\bul, L, \\Phi)\\Longrightarrow\\PD^d(A^\\bul, \\Phi)\\Longrightarrow\\WPD^d(A^\\bul). \\]\n\n\\item \\label{prop:HL:implication} {\\setlength{\\abovedisplayskip}{-1em}\n  \\[ \\HL^d(A^\\bul, L, \\Phi)\\iff \\begin{cases}\n    \\PD^d(A^\\bul, \\Phi)\\text{ and }\\\\\n    \\forall\\, k\\leq d\/2, \\,\\, L^{d-2k}\\colon A^k\\to A^{d-k}\\text{ is an isomorphism.}\n  \\end{cases} \\]}\n\n\\item \\label{prop:HR-sign} {\\setlength{\\abovedisplayskip}{-.5em}\n  \\[ \\HR^d(A^\\bul, L, \\Phi)\\iff\n  \\begin{cases}\n    \\WPD^d(A^\\bul) \\\\\n    \\forall k\\leq d\/2, \\text{ the restriction $(-1)^kQ_{L,\\Phi}\\rest{A^k_{\\prim}}$ is positive definite.}\n  \\end{cases} \\]}\n\n\\item \\label{prop:HR:automorphism}\n  For $S\\colon A^\\bul \\to A^\\bul$ an automorphism of degree 0,\n  \\begin{align*}\n  \\PD(A^\\bul,\\Phi) &\\iff \\PD(A^\\bul,\\Phi\\circ S), \\\\\n  \\HL(A^\\bul,Q)    &\\iff \\HL(A^\\bul,Q\\circ S),    \\\\\n  \\HR(A^\\bul,Q)    &\\iff \\HR(A^\\bul,Q\\circ S).\n  \\end{align*}\n\n\\item \\label{prop:HR:oplus}\n  For two graded $\\corps$-algebras $A^\\bul$ and $B^\\bul$, we have\n  \\begin{align*}\n  \\WPD(A^\\bul)       \\text{ and } \\WPD(B^\\bul)       &\\iff \\WPD(A^\\bul)       \\text{ and } \\WPD(A^\\bul\\oplus B^\\bul),                    \\\\\n  \\PD(A^\\bul,\\Phi_A) \\text{ and } \\PD(B^\\bul,\\Phi_B) &\\iff \\PD(A^\\bul,\\Phi_A) \\text{ and } \\PD(A^\\bul\\oplus B^\\bul,\\Phi_A\\oplus \\Phi_B), \\\\\n  \\HL(A^\\bul,Q_A)    \\text{ and } \\HL(B^\\bul,Q_B)    &\\iff \\HL(A^\\bul,Q_A)    \\text{ and } \\HL(A^\\bul\\oplus B^\\bul,Q_A\\oplus Q_B),       \\\\\n  \\HR(A^\\bul,Q_A)    \\text{ and } \\HR(B^\\bul,Q_B)    &\\iff \\HR(A^\\bul,Q_A)    \\text{ and } \\HR(A^\\bul\\oplus B^\\bul,Q_A\\oplus Q_B).\n  \\end{align*}\n\n\\item \\label{prop:HR:otimes}\n  For two graded $\\corps$-algebras $A^\\bul$ and $B^\\bul$, we have\n  \\begin{align*}\n  \\WPD(A^\\bul)\\text{ and }\\WPD(B^\\bul)             &\\Longrightarrow \\WPD(A^\\bul\\otimes B^\\bul), \\\\\n  \\PD(A^\\bul,\\Phi_A)\\text{ and }\\PD(B^\\bul,\\Phi_B) &\\Longrightarrow \\PD(A^\\bul\\otimes B^\\bul,\\Phi_A\\otimes \\Phi_B), \\\\\n  \\HL(A^\\bul,Q_A)\\text{ and }\\HL(B^\\bul,Q_B)       &\\Longrightarrow \\HL(A^\\bul\\otimes B^\\bul,Q_A\\otimes Q_B), \\\\\n  \\HR(A^\\bul,Q_A)\\text{ and }\\HR(B^\\bul,Q_B)       &\\Longrightarrow \\HR(A^\\bul\\otimes B^\\bul,Q_A\\otimes Q_B).\n  \\end{align*}\n\n\\item \\label{prop:HR:poincare_plus}\n  For two graded $\\corps$-algebras $A^\\bul$ and $B^\\bul$\n  \\begin{align*}\n  \\Phi_{\\deg_A}\\otimes\\Phi_{\\deg_B}   &= \\Phi_{\\deg_A\\otimes\\deg_B}, \\text{ and } \\\\\n  Q_{L_A,\\Phi_A}\\oplus Q_{L_B,\\Phi_B} &= Q_{L_A\\oplus L_B,\\Phi_A\\oplus\\Phi_B}.\n  \\end{align*}\n\n\\item \\label{prop:HR:otimes_lefschetz}\n  For two graded $\\corps$-algebras $A^\\bul$ and $B^\\bul$, we have\n  \\[ \\HR(A^\\bul,L_A,\\Phi_{A})\\text{ and }\\HR(B^\\bul,L_B,\\Phi_B)\\Longrightarrow\\HR(A^\\bul\\otimes B^\\bul, L_A\\otimes\\id+\\id\\otimes L_B,\\Phi_A\\otimes \\Phi_B). \\]\n\\end{enumerate}\n\\end{prop}\n\nMost of these statements are routine. We only give the proof of the first three and the very last statements.\n\\begin{proof}[Proof of~\\eqref{prop:HR:implication}]\nThe last implication is trivial. If $\\HL^d(A^\\bul, L, \\Phi)$ holds, then the bilinear form $Q=Q_{L, \\Phi}$ is non-degenerate. This implies that for each non-negative integer $k \\leq d\/2$, the map $L^{d-2k}$ is injective on $A^k$. By $\\WPD^d(A^\\bul)$, this implies that $L^{d-2k}$ induces an isomorphism between $A^k$ and $A^{d-k}$, and so, as $A^j$ and $A^k$ are orthogonal for distinct pair of integers $j$ and $k$, by our requirement, this implies that $\\Phi$ is non-degenerate and so we have $\\PD^d(A^\\bul, \\Phi)$.\n\nIt remains to prove the first implication. We prove by induction on $k$ that $Q_k$ is non-degenerate. The case $k=0$ is immediate by the assumption on the signature and the fact that $A^0 =\\corps\\cdot 1_{A^\\bul}$. Assume this holds for $k-1$ and $k \\leq d\/2$. Since $L$ is auto-adjoint, we have for any $a,b \\in A^{k-1}$,\n\\[Q_{k-1}(a,b) = Q_{k}(La,Lb)\\]\nwhich implies that $L\\colon A^{k-1} \\to A^k$ is injective, and $Q_k$ restricted to $L(A^{k-1})$ is non-degenerate and has signature that of $Q_{k-1}$. Since the difference between the signature of $Q_k$ and that of $Q_{k-1}$ is $(-1)^k \\bigl(\\dim(A^k) - \\dim(A^{k-1})\\bigr) = (-1)^k \\bigl(\\dim(A^k) - \\dim(LA^{k-1})\\bigr)$, it follows that $Q_k$ is non-degenerate.\n\\end{proof}\n\\begin{proof}[Proof of~\\eqref{prop:HL:implication}] The implication $\\Rightarrow$ follows from the proof of~\\eqref{prop:HR:implication}. The reverse implication is a consequence of our requirement that $A^j$ and $A^k$ are orthogonal for $\\Phi$ if $j$ and $k$ are distinct, which combined with the non-degeneracy of $\\Phi$ and the isomorphism between $A^k$ and $A^{d-k}$ induced by $L^{d-2k}$, implies that $Q_k$ is non-degenerate.\n\\end{proof}\n\\begin{proof}[Proof of~\\eqref{prop:HR-sign}] Let $Q = Q_{L,\\Phi}$. By~\\eqref{prop:HR:implication}, $\\HR^d(A^\\bul, L, \\Phi)$ implies $\\HL^d(A^\\bul, L, \\Phi)$. By Proposition~\\ref{prop:lefschetzdecomposition} below this leads to an orthogonal decomposition\n\\[ A^{k}=\\bigoplus_{i=0}^k L^{k-i}A^{i}_{\\prim} = A^k_\\prim \\oplus LA^{k-1}\\]\nwhere the restriction of $Q_k$ to $LA^{k-1}$ coincides with that of $Q_{k-1}$ on $A^{k-1}$. Proceeding by induction and using $\\HR^d(A^\\bul, L, \\Phi)$ which gives the signature of $Q_k$, we get the implication $\\Rightarrow$.\n\n\\medskip\n\nIt remains to prove the reverse implication $\\Leftarrow$. We proceed by induction and show that the bilinear form $Q_k$ is non-degenerate, that $L^{d-2k}\\colon A^{k}\\to A^{d-k}$ is an isomorphism, and that we have an orthogonal decomposition $A^k = A^k_\\prim \\bigoplus LA^{k-1}$. This shows that $L$ is an isometry with respect to $Q_{k-1}$ and $Q_k$ on the source and target, respectively, and the result on signature follows by the positivity of $(-1)^k Q_k$ on the primitive part combined with $\\dim A^k_\\prim = \\dim(A^{k}) -\\dim(A^{k-1})$.\n\nThe case $k=0$ comes from the assumption on the positivity of $Q_0$, since $A^0=A^0_\\prim$, which implies $L^d \\colon A^0 \\to A^d$ is injective, and the property $\\WPD^d(A^\\bul)$. Suppose the property holds for $k-1$ and that $k\\leq d\/2$. By the assumption of our induction, $L\\colon A^{k-1} \\to A^k$ is injective, and in addition, it preserves the bilinear forms $Q_{k-1}$ and $Q_k$. Moreover, $A^k_\\prim$ is orthogonal to $LA^{k-1}$. We now explain why an element $a \\in A^k $ which is orthogonal to $LA^{k-1}$ is in the primitive part $A^k_\\prim$: for such an element, and for an element $b\\in A^{k-1}$, we have\n\\[Q_k(a, Lb) = \\Phi(a, L^{d-2k+1}b) = \\Phi(L^{d-2k+1}a, b) =0.\\]\nSince $Q_{k-1}$ is non-degenerate, using $\\WPD^{d}$, we infer the bilinear form $\\Phi$ restricted to $A^{d-k+1} \\times A^{k-1}$ is non-degenerate. This implies that $L^{d-2k+1}a$ is zero and so $a\\in A^k_\\prim$.\n\n\\medskip\n\nWe thus get the orthogonal decomposition $A^k = A^k_\\prim \\oplus LA^{k-1}$. We infer that $Q_k$ is non-degenerate, and using $\\WPD^d(A^\\bul)$, we get the isomorphism $L^{d-2k} \\colon A^k \\to A^{d-k}$.\n\\end{proof}\n\n\\begin{proof}[Proof of \\eqref{prop:HR:otimes_lefschetz}]  By~\\eqref{prop:HR:implication} and Proposition~\\ref{prop:lefschetzdecomposition}, we can decompose each graded piece $A^k$ and $B^l$ for $k\\leq d_A\/2$ and $l \\leq d_B\/2$ into the orthogonal sum\n\\[ A^{k}=\\bigoplus_{i=0}^k L_A^{k-i}A^{i}_{\\prim} \\qquad B^{l}=\\bigoplus_{i=0}^l L_B^{l-i}B^{i}_{\\prim}\\]\nsuch that $Q_{A, k}$ and $Q_{B,j}$ are definite on $A^k_\\prim$ and $B^j_\\prim$, respectively. Taking an orthonormal basis of the primitive parts $A^k_\\prim$ and $B^j_\\prim$, and making a shift in the degree of graded pieces, by Proposition \\ref{prop:HR}, we can reduce to verifying the statement in the case where $A^\\bul =\\rquot{\\corps[\\x]}{\\x^{r+1}}$ and $B^\\bul = \\rquot{\\corps[\\y]}{\\y^{s+1}}$, for two non-negative integers $r$ and $s$, and $L_A$ and $L_B$ are multiplications by $\\x$ and $\\y$, respectively. The proof in this case can be obtained either by using Hodge-Riemann property for the projective  complex variety $\\mathbb C\\P^{r} \\times \\mathbb C\\P^s$, or by the direct argument given in~\\cite{BBFK}*{Proposition 5.7}, or still by the combinatorial argument given in~\\cite{McD}*{Lemma 2.2} and~\\cite{AHK}*{Lemma 7.8} based on the use of Gessel-Viennot-Lindstr\\\"om lemma~\\cites{GV, Lin}.\n\\end{proof}\n\n\\begin{prop}[Lefschetz decomposition]\\label{prop:lefschetzdecomposition}\nAssume that $\\HL^d(A^\\bul, L, \\Phi)$ holds. Then for $k\\leq\\frac{d}2$, we have the orthogonal decomposition \\[ A^{k}=\\bigoplus_{i=0}^k L^{k-i}A^{i}_{\\prim}, \\]\nwhere, for each $i\\in\\zint0k$, the map $L^{k-i}\\colon A^{i}_{\\prim} \\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} L^{k-i}A^{i}_{\\prim}$ is an isomorphism which preserves respective bilinear forms \\emph{($Q_i$ on $A^i$ and $Q_k$ on $A^{k}$)}.\n\\end{prop}\n\\begin{proof} First, we observe that for $i\\leq k$, the map $L^{k-i}$ induces an isomorphism between $A^{i}$ and its image $L^{k-i}A^{i}$ in\n$A^k$. This follows from the fact previously stated that for $a,b \\in A^{j}$,\n\\[Q_{j}(a,b) = Q_{j+1}(La,Lb)\\]\nwhich implies that $L\\colon A^{j} \\to A^{j+1}$ is injective provided that $j+1 \\leq d\/2$. Note that $A_\\prim$ is orthogonal to $LA^{k-1}$. Indeed for $a\\in A_\\prim$ and $b\\in A^{k-1}$, we have\n\\[\\Phi(a , L^{d-2k} Lb) = \\Phi(L^{d-2k+1}a , b)=0. \\]\nMoreover, an element $a \\in A^{k}$ orthogonal to $LA^{k-1}$ is necessarily in the primitive part, \\cf. the proof of~\\eqref{prop:HR-sign}.\nWe thus get the orthogonal decomposition\n\\[A^k = A^k_{\\prim} \\oplus L A^{k-1}.\\]\nProceeding by induction on $k$, this leads to the orthogonal decomposition\n\\[A^k = A_\\prim^k \\oplus L A^{k-1} = \\bigoplus_{i=0}^k L^{k-i}A^{i}_{\\prim}. \\qedhere \\]\n\\end{proof}\n\nThe following proposition is straightforward.\n\n\\begin{prop}\\label{prop:HRbis} The properties $\\HR$ and $\\HL$ are open conditions in $Q$ (and so in $\\Phi$ and $L$). The property $\\PD$ is open in $\\Phi$. Moreover, $\\HR$ is a closed condition in $Q$ when restricted to the space $\\Bigl\\{Q\\st\\HL^d(A^\\bul, Q)\\Bigr\\}$ consisting of those $Q$ which verify $\\HL$.\n\\end{prop}\n\n\n\n\\subsection{Chow rings of fans} For a simplicial fan $\\Sigma$ of dimension $d$ in a real vector space $V$, we denote by $A^\\bul(\\Sigma)$ its Chow ring with $\\corps$ coefficients. It is defined as follows.\n\nFor each ray $\\varrho \\in \\Sigma_1$, pick a vector $\\e_\\varrho$ which generates $\\varrho$.\nIf $\\Sigma$ is a rational fan with respect to a lattice, a natural choice for the generating vector is the primitive vector in the lattice which generates the ray.\n\nConsider the polynomial ring $\\corps[\\x_\\varrho]_{\\varrho\\in \\Sigma_1}$ with indeterminate variables $\\x_\\varrho$ associated to rays $\\varrho$ in $\\Sigma_1$. The Chow ring $A^\\bul(\\Sigma)$ is by definition the quotient ring\n\\[A^\\bul(\\Sigma) := \\rquot{\\corps[\\x_\\varrho]_{\\varrho\\in \\Sigma_1}}{\\bigl(\\I_1 + \\I_2\\bigr)}\\]\nwhere\n\\begin{itemize}\n\\item $\\I_1$ is the ideal generated by the products $\\x_{\\varrho_1}\\x_{\\varrho_2} \\dots \\x_{\\varrho_k}$ for any positive integer $k$ and rays $\\varrho_1, \\dots, \\varrho_k$ which do not form a cone in $\\Sigma$;\n\\item $\\I_2$ is the ideal generated by the homogeneous polynomials of degree one of the form $\\sum_{\\varrho\\in \\Sigma_1} f(\\e_\\varrho) \\x_\\varrho$ where $f \\in V^\\dual$ is a linear form on the vector space $V$, and $f(\\varrho)$ is the evaluation of $f$ at the generator $\\e_\\varrho$ of $\\varrho$.\n\\end{itemize}\n\nNote that since the ideal $\\I_1+\\I_2$ is homogeneous, the Chow ring inherits a graded ring structure. Moreover, the definition is well-posed even in the case of non-rational fans, in which case $\\corps=\\mathbb{R}$, as any two different choices of generating vectors for the rays lead to an isomorphism of the quotient rings. (This isomorphism sends the generator $\\x_\\varrho$ in the Chow ring defined with respect to the first generator set to $c_\\varrho \\x_\\varrho$ in the second, for a constant $c_\\varrho \\in \\mathbb{R}_{>0}$.)\n\n\\medskip\n\nFor each ray $\\varrho$ of $\\Sigma$, we denote by $x_\\varrho$ the image of $\\x_\\varrho$ in $A^1(\\Sigma)$.\\\\\nAny cone-wise linear function $f$ on $\\Sigma$ gives an element of $A^1(\\Sigma)$ defined as\n\\[\\sum_{\\varrho \\in \\Sigma_1} f(\\e_\\varrho) x_\\varrho.\\]\nIn fact, all elements of $A^1(\\Sigma)$ are of this form, and $A^1(\\Sigma)$ can be identified with the space of cone-wise linear functions on $\\Sigma$ modulo linear functions.\n\n\\medskip\n\nIn the case the fan $\\Sigma$ is rational and unimodular with respect to a lattice $N$ in $V$, one can define Chow rings with integral coefficients by choosing $\\e_\\varrho$ to be the primitive vector of $\\varrho$, for each ray $\\varrho$, and by requiring $f$ in the definition of $\\I_2$ to be integral.\nIn this case, we have the following characterization of the Chow ring, cf.~\\cites{Dan78, BDP90, Bri96, FS}.\n\\begin{thm}\nLet $\\Sigma$ be a unimodular fan, and denote by $\\P_\\Sigma$ the corresponding toric variety. The Chow ring $A^\\bullet(\\Sigma)$ is isomorphic to the Chow ring of $\\P_\\Sigma$.\n\\end{thm}\n\nIn the sequel, unless otherwise stated, we will be only considering Bergman fans, which are thus fans on the support of the Bergman fan of a matroid. Moreover, we suppose that all the matroids are simple which means they are loopless and do not have parallel elements. We call as before \\emph{Bergman support} the support of the Bergman fan of any matroid.\n\n\n\n\\subsection{Bergman supports are smooth} Let $\\Sigma$ be a Bergman fan. Recall that for any $\\sigma \\in \\Sigma$, we denote by $\\Sigma^\\sigma$ the star fan of $\\sigma$ which lives in the vector space $N^{\\sigma}_\\mathbb{R} = \\rquot{N_\\mathbb{R}}{N_{\\sigma, \\mathbb{R}}}$.\n\n\\begin{prop} \\label{prop:loc-smooth}For any Bergman fan $\\Sigma$, all the star fans $\\Sigma^\\sigma$ are Bergman. It follows that Bergman supports are smooth. Moreover, $\\Sigma$ is connected in codimension one.\n\\end{prop}\n\\begin{proof} See e.g.~\\cites{AP, Sha13a}.\n\\end{proof}\n\n\n\n\\subsection{The canonical element} The following proposition allows to define a degree map for the Chow ring of a Bergman fan.\n\n\\begin{prop} \\label{prop:canonical}Let $\\Sigma$ be a unimodular Bergman fan of dimension $d$. For each cone $\\sigma\\in\\Sigma_d$, the element\n\\[\\prod_{\\varrho\\prec \\sigma \\\\ \\dims \\varrho =1}x_\\varrho\\in A^d(\\Sigma)\\]\ndoes not depend on the choice of $\\sigma$.\n\\end{prop}\nWe call this element the \\emph{canonical element} of $A^d(\\Sigma)$ and denote it by $\\omega_\\Sigma$.\n\n\\begin{proof} By Proposition~\\ref{prop:loc-smooth}, $\\Sigma$ is smooth. Consider a face $\\tau$ of codimension one in a $d$-dimensional cone $\\sigma$ of $\\Sigma$. The star fan $\\Sigma^\\tau$ is Bergman. It has dimension one, and it follows by unimodularity that the primitive vectors $\\e_{\\eta\/\\tau}$ of the rays corresponding to $\\eta$ in $\\Sigma^\\tau$, for $\\eta \\ssupface \\tau$, form a circuit in $N^\\tau_\\mathbb{R}$: this means we have\n\\[\\sum_{\\eta \\ssupface \\tau} \\e_{\\eta\/\\tau} =0\\]\nand, moreover, this equation and its scalar multiples are the unique linear relations between these vectors. Combined with unimodularity, this implies that for any facet $\\zeta \\neq \\sigma$ in $\\Sigma$ which contains $\\tau$, we can find a linear function $f \\in {N^{\\tau}}^\\vee$ such that, viewing $f$ as an element of $N^\\vee$ which vanishes on $N_\\tau$, we have\n\\begin{itemize}\n\\item $f$ vanishes on $\\nvect_{\\eta\/\\tau}$ for any facet $\\eta \\ssupface \\tau$ distinct from $\\sigma$ and $\\zeta$, and\n\\item $f$ takes values $f(\\nvect_{\\sigma\/\\tau}) = -f(\\nvect_{\\zeta\/\\tau}) =1$.\n\\end{itemize}\nIf we denote by $\\rho_{\\eta\/\\tau}$ the unique ray of $\\eta\\in\\Sigma$ which is not in $\\tau$, we get\n\\[\\Bigr(\\sum_{\\eta \\ssupface \\tau} f(\\nvect_{\\eta\/\\tau}) x_{\\rho_{\\eta\/\\tau}}\\Bigl) \\prod_{\\varrho \\prec \\tau \\\\ \\dims \\varrho=1}x_\\varrho = \\Bigr(\\sum_{\\varrho\\in\\Sigma_1} f(\\varrho) x_{\\varrho}\\Bigl) \\prod_{\\varrho \\prec \\tau \\\\ \\dims \\varrho=1}x_\\varrho  = 0.\\]\nWe infer that\n\\[\\prod_{\\varrho \\prec \\sigma \\\\ \\dims\\varrho=1}x_\\varrho  = \\prod_{\\varrho \\prec \\zeta \\\\ \\dims \\varrho=1}x_\\varrho. \\]\n\nThis shows that for any two facets $\\sigma$ and $\\zeta$ which share a face of dimension $d-1$, the two associated elements\n$\\prod_{\\varrho\\prec \\sigma \\\\ \\dims \\varrho =1}x_\\varrho$ and $\\prod_{\\varrho\\prec \\zeta \\\\ \\dims \\varrho =1}x_\\varrho$ coincide in $A^d(\\Sigma)$. Since $\\Sigma$ is connected in codimension one, this proves the proposition.\n\\end{proof}\n\n\\begin{prop}\nLet $\\Sigma$ be a unimodular Bergman fan. Then, for each $k\\in\\{0,\\dots,d\\}$, $A^k(\\Sigma)$ is generated by square-free monomials.\n\\end{prop}\n\\begin{proof}\nTo prove this statement, one can proceed by a lexicographic induction and use the relations defining the Chow ring in order to replace monomials which have exponents larger than one by a linear combination of square-free monomials, as in~\\cite{AHK} and \\cite{Ami}.\n\\end{proof}\n\n\\begin{cor}\\label{prop:non-vanishing} Let $\\Sigma$ be a unimodular Bergman fan. Then we have $\\omega_\\Sigma \\neq 0$ and $A^d(\\Sigma)$ is generated by $\\omega_\\Sigma$.\n\\end{cor}\n\\begin{proof} This follows from the fact that $A^d(\\Sigma)$ is generated by square-free monomials associated to top-dimensional cones, which by Poincar\\'e duality $A^0(\\Sigma) \\simeq A^{d}(\\Sigma)^\\dual$ and Proposition \\ref{prop:canonical} ensures the non-vanishing of $\\omega_\\Sigma$.\n\\end{proof}\n\n\\medskip\n\nLet $\\deg\\colon A^d(\\Sigma)\\to\\corps$ be the linear map which takes value 1 on the canonical element. This degree map and the corresponding bilinear map $\\Phi_{\\deg}$ is systematically used below, so we use our convention and omit the mention of $\\Phi$ and $\\deg$ in $\\HL$ and $\\HR$.\n\n\n\n\\subsection{Chow rings of products} Let $\\Sigma$ and $\\Sigma'$ be two Bergman fans. By Proposition~\\ref{prop:productBergman} the product $\\Sigma \\times \\Sigma'$ is Bergman. We have the following proposition for the Chow ring of the product fan whose proof is straightforward and is thus omitted.\n\n\\begin{prop} \\label{prop:cartesian_product}\nLet $\\Sigma$ and $\\Sigma'$ be two fans. There exists a natural isomorphism of rings\n\\[ A^\\bul(\\Sigma\\times\\Sigma')\\simeq A^\\bul(\\Sigma)\\otimes A^\\bul(\\Sigma'). \\]\nMoreover, under this isomorphism, one has\n\\begin{gather*}\n\\omega_{\\Sigma\\times\\Sigma'}=\\omega_\\Sigma\\otimes\\omega_{\\Sigma'}, \\\\\n\\deg_{\\Sigma\\times\\Sigma'}=\\deg_\\Sigma\\otimes\\deg_{\\Sigma'}.\n\\end{gather*}\n\\end{prop}\n\n\n\n\\subsection{Restriction and Gysin maps} For a pair of cones $\\tau \\prec \\sigma$ in $\\Sigma$, we define the restriction and Gysin maps $\\i^*_{\\tau \\prec \\sigma}$ and $\\gys_{\\sigma \\succ \\tau}$ between the Chow rings of $\\Sigma^\\tau$ and $\\Sigma^\\sigma$.\n\n\\medskip\n\nThe restriction map\n\\[\\i^*_{\\tau \\prec \\sigma}\\colon A^\\bul(\\Sigma^\\tau) \\to A^\\bul(\\Sigma^\\sigma)\\]\nis a graded $\\corps$-algebra homomorphism given on generating sets\n\\[\\i^*_{\\tau \\prec \\sigma}\\colon A^1(\\Sigma^\\tau) \\to A^1(\\Sigma^\\sigma)\\]\nby\n\\[\n\\forall \\, \\rho \\textrm{ ray in } \\Sigma^\\tau, \\qquad \\i^*_{\\tau \\prec \\sigma} (x_\\rho) =\n\\begin{cases}\n  x_\\rho & \\qquad  \\textrm{if $\\sigma+\\rho$ is a cone $\\zeta \\succ \\sigma$}, \\\\\n  -\\sum_{\\varrho \\in \\Sigma^\\sigma_1} f(\\e_\\varrho) x_\\varrho & \\qquad \\textrm{if $\\rho \\in \\sigma$} \\\\\n  0 & \\qquad \\textrm{otherwise},\n\\end{cases}\n\\]\nwhere in the second equality, $f$ is any linear function on $N^\\tau$, viewed naturally in $N^\\vee$, which takes value $1$ on $\\e_\\rho$ and value zero on all the other rays of $\\sigma$, and the sum is over all the rays $\\varrho$ of $\\Sigma^\\sigma$. Note that any two such choices of $f$ and $f'$ differ by a linear function on $N^\\sigma$, and so the element $\\sum_{\\varrho \\in \\Sigma^\\sigma} f(\\e_\\varrho) x_\\varrho$ in $A^1(\\Sigma^\\sigma)$ does not depend on the choice of $f$.\n\n\\begin{remark} For a pair of cones $\\tau \\prec \\sigma$ as above, we get an inclusion of canonical compactifications $\\i \\colon \\comp\\Sigma^\\sigma \\hookrightarrow \\comp\\Sigma^{\\tau}$. The map $\\i^*_{\\tau \\prec \\sigma}\\colon A^k(\\Sigma^\\tau) \\to A^k(\\Sigma^\\sigma)$ coincides with the restriction map\n\\[\\i^* \\colon H^{k,k}_\\trop(\\comp\\Sigma^{\\tau}) \\longrightarrow H^{k,k}_\\trop(\\comp\\Sigma^{\\sigma}) \\]\nvia the isomorphism between the Chow and tropical cohomology groups given in Theorem~\\ref{thm:HI}.\n\\end{remark}\n\nThe Gysin map is the $\\corps$-linear morphism\n\\[ \\gys_{\\sigma \\succ \\tau} \\colon A^{\\bul}(\\Sigma^\\sigma) \\longrightarrow A^{\\bul+ \\dims \\sigma -\\dims \\tau}(\\Sigma^\\tau) \\]\ndefined as follows. Let $r = \\dims \\sigma -\\dims \\tau$, and denote by $\\rho_1, \\dots, \\rho_r$ the rays of $\\sigma$ which are not in $\\tau$.\n\nConsider the $\\corps$-linear map\n\\[\\corps[\\x_\\varrho]_{\\substack{\\varrho \\in \\Sigma_1^\\sigma}} \\longrightarrow \\corps[\\x_\\varrho]_{\\substack{\\varrho \\in \\Sigma^\\tau_1}}\\]\ndefined by multiplication by $\\x_{\\rho_1}\\x_{\\rho_2} \\dots \\x_{\\rho_r}$. Obviously, it sends an element of the ideal $\\I_2$ in the source to an element of the ideal $\\I_2$ in the target. Moreover, any linear function on $N^\\sigma$ defines a linear function on $N^\\tau$ via the projection\n\\[N^\\tau = \\rquot{N}{N_\\tau} \\longrightarrow N^\\sigma = \\rquot{N}{N_{\\sigma}}.\\]\nThis shows that the elements of $\\I_1$ in the source are sent to elements of $\\I_1$ in the target as well. Passing to the quotient, we get a $\\corps$-linear map\n\\[ \\gys_{\\sigma \\succ \\tau} \\colon A^{k}(\\Sigma^\\sigma) \\longrightarrow A^{k+ \\dims \\sigma -\\dims \\tau}(\\Sigma^\\tau). \\]\n\\begin{remark} For a pair of cones $\\tau \\prec \\sigma$ as above, from the inclusion of canonical compactifications $\\i \\colon \\comp\\Sigma^\\sigma \\hookrightarrow \\comp\\Sigma^{\\tau}$, we get a restriction map\n\\[\\i^*\\colon H^{d-\\dims \\sigma -k,d-\\dims \\sigma - k}_\\trop(\\comp\\Sigma^{\\tau}) \\longrightarrow H^{d-\\dims \\sigma-k,d-\\dims \\sigma -k}_\\trop(\\comp\\Sigma^{\\sigma}) \\]\nwhich by Poincar\\'e duality on both sides, gives the Gysin map\n\\[\\gys_{\\sigma>\\tau} \\colon H^{k,k}_\\trop(\\comp\\Sigma^{\\sigma})  \\longrightarrow H^{k+\\dims \\sigma -\\dims\\tau,k +\\dims \\sigma -\\dims \\tau}_\\trop(\\comp\\Sigma^{\\tau}).\\]\nThis map coincides with the Gysin map between Chow groups defined in the preceding paragraph, via the isomorphism of Chow and tropical cohomology given in Theorem~\\ref{thm:HI}.\n\\end{remark}\n\nThe following proposition gathers some basic properties of the restriction and Gysin maps.\n\\begin{prop} \\label{lem:i_gys_basic_properties-local}\nLet $\\tau\\ssubface\\sigma$ be a pair of faces, and let $x\\in A^\\bul(\\Sigma^\\tau)$ and $y\\in A^\\bul(\\Sigma^\\sigma)$. Denote by $\\rho_{\\sigma\/\\tau}$ the unique ray associated to $\\sigma$ in $\\Sigma^\\tau$, and by $x_{\\sigma\/\\tau}$ the associated element of $A^1(\\Sigma^\\tau)$. The following properties hold.\n\\begin{gather}\n\\text{$\\i^*_{\\tau\\ssubface\\sigma}$ is a surjective ring homomorphism.} \\label{eqn:i_surjective_homeo-local} \\\\\n\\gys_{\\sigma\\ssupface\\tau}\\circ\\i^*_{\\tau\\ssubface \\sigma}(x)=x_{\\sigma\/\\tau}\\cdot x. \\label{eqn:gys_circ_i-local} \\\\\n\\gys_{\\sigma\\ssupface\\tau}(\\i^*_{\\tau\\ssubface\\sigma}(x)\\cdot y)=x\\cdot\\gys_{\\sigma \\ssupface\\tau}(y).  \\label{eqn:gys_i_simplification-local}\n\\end{gather}\nMoreover, if $\\deg_\\tau\\colon A^{d-\\dims\\tau}(\\Sigma^\\tau)\\to\\corps$ and $\\deg_\\sigma\\colon A^{d-\\dims\\sigma}(\\Sigma^\\sigma)\\to\\corps$ are the corresponding degrees maps, then\n\\begin{equation} \\label{eqn:deg_circ_gys-local}\n\\deg_\\sigma=\\deg_\\tau\\circ\\gys_{\\sigma\\ssupface\\tau}.\n\\end{equation}\nFinally, $\\gys_{\\sigma\\ssupface\\tau}$ and $\\i^*_{\\tau\\ssubface\\sigma}$ are dual in the sense that\n\\begin{equation} \\label{eqn:i_gys_dual-local}\n\\deg_\\tau(x\\cdot\\gys_{\\sigma\\ssupface\\tau}(y))=\\deg_\\sigma(\\i^*_{\\tau\\ssubface\\sigma}(x)\\cdot y).\n\\end{equation}\n\\end{prop}\n\n\\begin{proof} In order to simplify the presentation, we drop the indices of $\\gys$ and $\\i^*$. Properties \\eqref{eqn:i_surjective_homeo-local} and \\eqref{eqn:gys_circ_i-local} follow directly from the definitions. From Equation \\eqref{eqn:gys_circ_i-local}, we can deduce Equation \\eqref{eqn:gys_i_simplification-local} by the following calculation. Let $\\~y$ be a preimage of $y$ by $\\i^*$. Then,\n\\[ \\gys(\\i^*(x)\\cdot y)=\\gys(\\i^*(x\\cdot \\~y))=x_{\\delta\/\\gamma}\\cdot x\\cdot\\~y=x\\cdot\\gys\\circ\\i^*(\\~y)=x\\cdot\\gys(y). \\]\nFor Equation \\eqref{eqn:deg_circ_gys-local}, let $\\eta$ be a maximal cone of $\\Sigma^\\sigma$. Let $\\~\\eta$ be the corresponding cone containing $\\rho_{\\sigma\/\\tau}$ in $\\Sigma^\\tau$. The cone $\\~\\eta$ is maximal in $\\Sigma^\\tau$. We have the respective corresponding generators $x_{\\~\\eta}\\in A^{d-\\dims\\tau}(\\Sigma^\\tau)$ and $x_\\eta\\in A^{d-\\dims\\sigma}(\\Sigma^\\sigma)$. By definition of the  degree maps, $\\deg_\\tau(x_{\\~\\eta})=\\deg_\\sigma(x_\\eta)$. Using the definition of $\\gys$, we get that $x_{\\~\\eta}=\\gys(x_\\eta)$. Thus, we can conclude that\n\\[ \\deg_\\sigma = \\deg_\\tau\\circ\\gys. \\]\nFinally, we get Equation \\eqref{eqn:i_gys_dual-local}:\n\\[ \\deg_\\tau(x\\cdot\\gys(y))=\\deg_\\tau(\\gys(\\i^*(x)\\cdot y))=\\deg_\\sigma(\\i^*(x)\\cdot y). \\qedhere \\]\n\\end{proof}\n\n\n\n\\subsection{Primitive parts of the Chow ring} Let $\\Sigma$ be a Bergman fan of dimension $d$.\nFor an element $\\ell \\in A^1(\\Sigma)$ and any non-negative integer number $k \\leq \\frac d2$, the primitive part $A^k_{\\prim, \\ell}(\\Sigma)$ of $A^k(\\Sigma)$ is the kernel of the Lefschetz operator given by multiplication with $\\ell^{d-2k+1}$\n\\[\\ell^{d-2k+1}\\cdot - \\colon A^k(\\Sigma) \\longrightarrow A^{d-k+1}(\\Sigma).\\]\nLet $Q$ be the bilinear form defined on $A^k$, for $k \\leq \\frac{d}2$, by\n\\[ \\forall \\, a,b\\in A^{k}(\\Sigma), \\qquad Q (a,b) = \\deg(\\ell^{d-2k}ab).\\]\nNote that by Propositions~\\ref{prop:HR} and~\\ref{prop:lefschetzdecomposition}, we have the following properties.\n\\begin{itemize}\n\\item $\\HL(\\Sigma, \\ell)$ is equivalent to the following: For each $k \\leq \\frac{d}2$, the map\n\\[\\ell^{d-2k} \\cdot - \\colon A^{k}(\\Sigma) \\to A^{d-k}(\\Sigma)\\]\nis an isomorphism.\n\n\\item $\\HR(\\Sigma, \\ell)$ is equivalent to the following: The bilinear form $(-1)^k Q(\\,\\cdot\\,,\\cdot\\,)$ restricted to the primitive part $A^{k}_{\\prim}(\\Sigma)$ is positive definite.\n\n\\item Assume that $\\HL(\\Sigma, \\ell)$ holds. Then for $k\\leq\\frac{d}2$, we have the Lefschetz decomposition \\[ A^{k}(\\Sigma)=\\bigoplus_{i=0}^k \\ell^{k-i}A^{i}_{\\prim}(\\Sigma). \\]\n\\end{itemize}\n\n\n\n\\subsection{Hodge-Riemann for star fans of rays implies Hard Lefschetz} Let $\\ell \\in A^1(\\Sigma)$. We assume that $\\ell$ has a representative in $A^1(\\Sigma)$ with strictly positive coefficients, i.e.,\n\\[\\ell =\\sum_{\\varrho \\in \\Sigma_1} c_\\varrho x_\\varrho\\]\nfor scalars $c_\\varrho >0$ in $\\corps$. For each $\\varrho \\in \\Sigma_1$, define $\\ell^\\varrho = \\i^*_{\\conezero \\prec \\varrho}(\\ell) \\in A^1(\\Sigma^\\varrho)$.\n\n\\medskip\n\nThe following is well-known, see e.g.~\\cite{CM05} or~\\cite{AHK}*{Proposition 7.15}.\n\\begin{prop} \\label{prop:local_HR}\nIf $\\HR(\\Sigma^\\varrho, \\ell^\\varrho)$ holds for all rays $\\varrho\\in\\Sigma_1$, then we have $\\HL(\\Sigma, \\ell)$.\n\\end{prop}\n\n\\begin{proof} Let $k \\leq \\frac d2$ be a non-negative integer where $d$ denotes the dimension of $\\Sigma$. We need to prove that the\nbilinear form $Q_\\ell$ defined by\n\\[\\forall \\, a, b \\in A^{k}(\\Sigma), \\qquad Q_\\ell (a, b)=  \\deg(\\ell^{d-2k}ab)\\]\nis non-degenerate. By Poincar\\'e duality for $A^\\bul(\\Sigma)$, Theorem \\ref{thm:pd}, this is equivalent to showing the multiplication map\n\\[\\ell^{d-2k}\\cdot - \\colon A^{k}(\\Sigma) \\longrightarrow A^{d-k}(\\Sigma)\\]\nis injective. Let $a \\in A^k(\\Sigma)$ such that $\\ell^{d-2k}\\cdot a =0$. We have to show that $a=0$.\n\n\\medskip\n\nThere is nothing to prove if $k=d\/2$, so assume $2k < d$. For each $\\varrho \\in \\Sigma_1$, define $a_\\varrho :=\\i^*_{\\conezero \\prec \\varrho}(a)$.\nIt follows that\n\\[(\\ell^\\varrho)^{d-2k} \\cdot a_\\varrho = \\i^*_{\\conezero \\prec \\varrho}(\\ell^{d-2k}a) =0,\\]\nand so $a_\\varrho \\in A^{k}_{\\prim, \\ell^\\varrho}(\\Sigma^\\varrho)$ for each ray $\\varrho \\in \\Sigma_1$.\n\n\\medskip\n\nUsing now Proposition~\\ref{lem:i_gys_basic_properties-local}, for each $\\varrho\\in \\Sigma_1$, we get\n\\[\\deg_\\varrho\\bigl((\\ell^\\varrho)^{d-2k-1}\\cdot a_\\varrho \\cdot a_{\\varrho}\\bigr) = \\deg_\\varrho\\bigl(\\i^*_{\\conezero \\prec \\varrho}(\\ell^{d-2k-1}\\cdot a) \\cdot a_{\\varrho}\\bigr) = \\deg\\bigl(\\ell^{d-2k-1}\\cdot a \\cdot x_\\varrho \\cdot a\\bigr).\\]\n\nWe infer that\n\\[\\sum_{\\varrho \\in \\Sigma_1} c_\\varrho \\deg_\\varrho\\bigl((\\ell^\\varrho)^{d-2k-1}\\cdot a_\\varrho \\cdot a_{\\varrho}\\bigr) = \\deg\\bigl(\\ell^{d-2k-1}\\cdot a \\cdot \\bigl(\\,\\sum_{\\varrho }c_\\varrho x_\\varrho\\bigr) \\cdot a\\bigr) = \\deg(\\ell^{d-2k}\\cdot a\\cdot a)=0.\\]\n\nBy $\\HR(\\Sigma^\\varrho,\\ell^\\varrho)$, since $a_\\varrho\\in A^{k}_{\\prim, \\ell^\\varrho}(\\Sigma^\\varrho)$, we have $(-1)^k \\deg_\\varrho\\bigl((\\ell^\\varrho)^{d-2k-1}\\cdot a_\\varrho \\cdot a_{\\varrho}\\bigr) \\geq 0$ with equality if and only if $a_\\varrho =0$. Since $c_\\varrho >0$ for all $\\varrho$, we conclude that $a_\\varrho =0$ for all $\\varrho \\in \\Sigma_1$.\n\n\\medskip\n\nApplying Proposition~\\ref{lem:i_gys_basic_properties-local} once more, we infer that\n\\[x_\\varrho a =  \\gys_{\\varrho \\succ \\conezero}\\circ \\i^*_{\\conezero \\prec \\varrho} (a) = \\gys_{\\varrho \\succ \\conezero}(a_\\varrho) =0.\\]\n\nBy Poincar\\'e duality for $A^\\bul(\\Sigma)$, and since the elements $x_\\varrho$ generate the Chow ring, this implies that $a=0$, and the proposition follows.\n\\end{proof}\n\n\n\n\\subsection{Isomorphism of Chow groups under tropical modification} In this section, we prove that tropical modifications as defined in Section~\\ref{sec:modification} preserve the Chow groups. This will be served as the basis of our induction process to prove $\\HR$ and $\\HL$ properties for Bergman fans.\n\n\\begin{prop} \\label{prop:modification} Let $\\Ma$ be a simple matroid of rank $d+1$ on a ground set $E$, and let $a$ be a proper element of $E$. Let $\\Delta'$ be a unimodular Bergman fan with support $\\supp{\\Sigma_{\\Ma\\contr a}}$, and let $\\Sigma'$ be a unimodular Bergman fan with support $\\supp{\\Sigma_{\\Ma\\del a}}$ which contains $\\Delta'$ as a subfan. Let $\\Sigma$ be the unimodular fan obtained by the tropical modification of $\\Sigma'$ along $\\Delta'$. Then we have\n\\[A^\\bul(\\Sigma) \\simeq A^\\bul(\\Sigma').\\]\n\\end{prop}\n\n\\begin{proof} We follow the notation of Section~\\ref{sec:modification}. So $\\Sigma$ is embedded in $N_\\mathbb{R}$ for $N = \\rquot{\\mathbb{Z}^E}{\\mathbb{Z} \\e_E}$, and $(\\e_b)$ for $b\\in E$ represents the standard basis of $\\mathbb{Z}^E$. We denote by $M = N^\\vee$ the dual lattice. We also denote by $N^a$ the lattice $\\rquot{\\mathbb{Z}^{E - a}}{\\mathbb{Z} \\e_{_{E-a}}} = \\rquot{N}{\\mathbb{Z} \\e_a}$ and $M^a$ the corresponding dual lattice. Thus, we have $\\Sigma'\\subset N^a_\\mathbb{R}$. The tropical modification induces an injective map of fans $\\Gamma\\colon \\Sigma'\\to\\Sigma$ which preserves the fan structure. We denote by $\\rho_a$ the ray in $\\Sigma$ associated to $a$; note that with our convention, we have $\\e_{\\rho_a} =\\e_a$. The ray $\\rho_a$ is the only one which is not in $\\Gamma(\\Sigma')$.\n\n\\medskip\n\nLet $h$ be a linear form in $M_\\corps$ which takes value one on $\\e_a$. Then $h$ induces a piecewise linear function $f := \\Gamma^{-1}\\circ h\\rest{\\Gamma(\\Sigma')}$ on $\\Sigma'$ and we have $\\div(f) = -\\Delta'$, in the sense that the sum of slopes of $f$ around all codimension one faces $\\sigma\\in\\Sigma'_{d-1}\\setminus\\Delta'_{d-1}$ is zero while the sum of slopes is $-1$ for all $\\sigma\\in\\Delta'_{d-1}$.\n\nConsider the surjective $\\corps$-algebra homomorphism\n\\[\\phi\\colon \\corps[\\x_\\varrho]_{\\varrho\\in \\Sigma_1} \\longrightarrow \\corps[\\x_\\varrho]_{\\varrho\\in \\Sigma'_1}\\]\nwhich on the level of generators is defined by sending $\\x_{\\rho_a}$ to $- \\sum_{\\varrho \\in\\Sigma_1 - \\rho_a} f(\\e_\\varrho)\\x_\\varrho$, and which maps $\\x_\\varrho$ to $\\x_{\\Gamma^{-1}(\\varrho)}$ for rays $\\varrho$ of $\\Sigma_1$ different from $\\rho_a$. We will show that $\\phi$ induces an isomorphism between $A^\\bul(\\Sigma)$ and $A^\\bul(\\Sigma')$. For this, we first show that we have $\\phi(\\I_2) \\subseteq \\I_2$ and $\\phi(\\I_1) \\subseteq \\I_1+\\I_2$, which allows to pass to the quotient and to get a morphism of $\\corps$-algebra between the two Chow rings.\n\n\\medskip\n\nTo show that $\\phi(\\I_2) \\subseteq \\I_2$, consider a linear form $l$ on $N_\\corps$. Let $l' := l - l(\\e_a)h$. We have $l'(\\e_a)=0$ and so $l'$ gives a linear form on $N^a_\\corps$. We have\n\\[ \\phi(\\sum_{\\varrho\\in\\Sigma_1} l(\\e_\\varrho)\\x_\\varrho) = \\sum_{\\varrho\\in \\Sigma_1-\\rho_a} l(\\e_\\varrho)\\x_\\varrho+l(\\e_a)\\phi(\\x_{\\rho_a})=\\sum_{\\varrho\\in \\Sigma'_1} l'(\\e_\\varrho)x_\\varrho. \\]\nThis shows that\n\\[\\phi(\\I_2) \\subseteq \\I_2.\\]\n\nWe now consider the image of $\\I_1$. Consider a collection of distinct rays $\\varrho_0, \\dots, \\varrho_k$, $k\\in \\mathbb N$, and suppose they are not comparable, which translates to $\\x_{\\varrho_0} \\dots \\x_{\\varrho_k} \\in \\I_1$.\n\n\\medskip\n\nIf $\\varrho_0, \\dots, \\varrho_k \\neq \\rho_a$, then they all belong to $\\Sigma'$ and we have $\\phi(\\x_{\\varrho_0} \\dots \\x_{\\varrho_k}) = \\x_{\\varrho_0} \\dots \\x_{\\varrho_k} \\in \\I_1$.\n\n\\medskip\n\nOtherwise, one of the rays, say $\\varrho_0$, is equal to $\\rho_a$. Two cases can happen:\n\n\\begin{enumerate}\n\\item \\label{prop:topmodif_case1} Either, $\\varrho_1, \\dots, \\varrho_k$ belong to $\\Delta'$,\n\\item \\label{prop:tropmodif_case2} Or, one of the rays, say $\\varrho_k$, does not belong to $\\Delta'$.\n\\end{enumerate}\n\nIn the first case, since $\\rho_a, \\varrho_1, \\dots, \\varrho_k$ are not comparable, we deduce that the rays $\\varrho_1, \\dots, \\varrho_k$ are not comparable in $\\Delta'$, and again, we get\n\\[\\phi(\\x_{\\varrho_0}\\x_{\\varrho_1} \\dots \\x_{\\varrho_k}) = \\phi(\\x_{\\varrho_0}) \\x_{\\varrho_1} \\dots \\x_{\\varrho_k} \\in \\I_1.\n\\]\nSuppose we are in case~\\ref{prop:tropmodif_case2}, so $\\varrho_k \\not \\in \\Delta_1'$. It follows from Proposition~\\ref{prop:voisinage} below applied to $\\varrho =\\varrho_k$ that there exists a neighborhood of $\\e_{\\varrho_k}$ in $\\Sigma$ which is entirely included in a hyperplane $H$ in $N_\\corps$ which does not contain $\\rho_a$. From this we infer that any ray comparable with $\\varrho_k$ should belong to $H$. Moreover, the projection map $N \\to N^a$ induces an isomorphism $H\\to N^a_\\corps$. It follows that $h\\rest H$ induces a linear function $l$ on $N^a_\\corps$, which coincides with $h$ on all rays $\\xi$ which are comparable with $\\varrho_k$.\\\\\nTherefore,\n\\begin{align*}\n\\phi(\\x_{\\rho_a}\\x_{\\varrho_1} \\dots \\x_{\\varrho_k})\n&= - \\sum_{\\xi\\in\\Sigma_1-\\rho_a} h(\\e_\\xi)\\x_\\xi \\x_{\\varrho_1} \\dots \\x_{\\varrho_k} \\\\\n&=- \\sum_{\\xi\\in\\Sigma'_1}l(\\e_\\xi) \\x_\\xi \\x_{\\varrho_1} \\dots \\x_{\\varrho_k}\\in \\I_2.\n\\end{align*}\n\nFrom the discussion above, we obtain a surjective $\\corps$-algebra morphism\n\\[\\phi\\colon A^\\bul(\\Sigma) \\to A^\\bul(\\Sigma').\\]\nThis map is alos injective. Indeed, a reasoning similar to the above shows that the injection \\[\\corps[\\x_\\varrho]_{\\varrho\\in \\Sigma'_1} \\hookrightarrow \\corps[\\x_\\varrho]_{\\varrho\\in \\Sigma_1}\\]\npasses to the quotient and provides a morphism\n\\[A^\\bul(\\Sigma') \\to A^\\bul(\\Sigma)\\]\nwhich is clearly inverse to $\\phi$.\n\\end{proof}\n\n\\begin{prop} \\label{prop:voisinage} Notations as in the previous proposition, let $\\rho_a$ be the unique ray in $\\Sigma\\setminus \\Gamma(\\Sigma')$ and let $N = \\rquot{\\mathbb{Z}^{E}}{\\mathbb{Z} {\\e_{E}}}$. Let $\\varrho$ be a ray in $\\Gamma(\\Sigma') \\setminus \\Gamma(\\Delta')$. There exists a hyperplane $H$ in $N_\\corps$ which does not contain $\\rho_a$ such that there exists a neighborhood $U$ of $\\e_\\varrho$ in $\\Sigma$ which is entirely included in $H_\\mathbb{R}$.\n\\end{prop}\n\n\\begin{proof} Let as before $N^a$ be the lattice $\\rquot{\\mathbb{Z}^{E - a}}{\\mathbb{Z} \\e_{_{E-a}}} = \\rquot{N}{\\mathbb{Z} \\e_a}$ so that both the fans $\\Sigma'$ and $\\Delta'$ live in $N^a_\\mathbb{R}$.\n\nLet $T$ be the tangent space to $\\Sigma$ at $\\e_\\varrho$. Viewing $\\e_\\varrho$ in the Bergman fan $\\Sigma_{\\Ma}$ of the matroid $\\Ma$, there exists a flag of flats $\\mathscr{F}$ of $\\Ma$ consisting of proper flats $\\emptyset \\neq F_1 \\subsetneq F_2 \\subsetneq \\dots \\subsetneq F_i \\neq E$ such that $\\e_\\varrho$ lies in the relative interior of $\\sigma_\\mathscr{F}$ in $\\Sigma_{\\Ma}$. Note that since $\\varrho\\in\\Gamma(\\Sigma')$, the flats $F_j$ are in fact elements of $\\Cl(\\mu\\del a)$; in particular, they do not contain $a$. The linear subspace $T \\subseteq N_\\mathbb{R}$ is then generated by the vectors $\\e_F$ for $F$ a flat of $\\Ma$ which is compatible with $\\mathscr{F}$, in the sense that $\\mathscr{F} \\cup \\{F\\}$ forms again a flag of flats of $\\Ma$ ($F$ may belong to $\\mathscr{F}$).\n\nSince $\\varrho$ is not a ray of $\\Gamma(\\Delta')$, the cone $\\sigma_\\mathscr{F}$ is not in $\\supp{\\Gamma(\\Delta')} =\\supp{\\Gamma(\\Sigma_{\\Ma\\contr a})}$. This means there is an element $F_j \\in \\mathscr{F}$ which is not a flat of $\\Ma \\contr a$. By characterization of flats of $\\Ma \\del a$ and $\\Ma \\contr a$ given in Section~\\ref{sec:modification}, we infer that $F_j \\in \\Cl(\\Ma)$ but $F_j+a \\notin \\Cl(\\Ma)$. In particular, the closure $\\cl(F_j+a)$ in $\\Ma$ of $F_j+a$ contains an element $b \\in E \\setminus (F_j+a)$.\n\n\\medskip\n\nNow, for any flat $F$ in $\\Ma$ which is compatible with $\\mathscr{F}$, we have\n\\begin{itemize}\n\\item Either, $F$ contains both $a$ and $b$.\n\\item Or, $F$ contains none of $a$ and $b$.\n\\end{itemize}\nIndeed, if $F \\subseteq F_j$, then $F$ contains none of $a,b$. Otherwise, $F_j \\subseteq F$, and in this case, if $F$ contains one of $a$ or $b$, then it must contain the other by the property that $\\cl(F_j+a) =\\cl(F_j+b)$.\n\n\\medskip\n\nWe infer that for any such $F$, we have $\\e_F \\in H_\\mathbb{Z}:=\\ker(\\e_a^\\dual -\\e_b^\\dual)$, where $\\e_a^\\dual -\\e_b^\\dual \\in M = N^\\vee$. This implies $T \\subseteq H$ for $H = H_\\mathbb{Z} \\otimes_\\mathbb{Z} \\corps$. Obviously, $\\e_a \\notin H$ and the result follows.\n\\end{proof}\n\n\n\n\\subsection{Keel's lemma} \\label{sec:Keel} Let $\\Sigma$ be a unimodular fan and let $\\sigma$ be a cone in $\\Sigma$. Let $\\Sigma'$ be the star subdivision of $\\Sigma$ obtained by star subdividing $\\sigma$. Denote by $\\rho$ the new ray in $\\Sigma'$. In this paper we only consider barycentric star subdivisions of rational fans. This means that $\\rho=\\mathbb{R}_+(\\e_1+\\dots+\\e_k)$ where $\\e_1, \\dots, \\e_k$ are the primitive vectors of the rays of $\\sigma$.\n\n\\begin{thm}[Keel's lemma] \\label{thm:keel}\nLet $J$ be the kernel of the surjective map $\\i^*_{\\conezero\\prec\\sigma}\\colon A^\\bul(\\Sigma)\\to A^\\bul(\\Sigma^\\sigma)$ and let\n\\[ P(T):=\\prod_{\\varrho\\prec \\sigma \\\\ \\dims\\varrho=1}(x_\\varrho+T). \\]\nWe have\n\\[ A^\\bul(\\Sigma')\\simeq \\rquot{A^\\bul(\\Sigma)[T]}{(JT+P(T))}. \\]\nThe isomorphism is given by the map\n\\[\\chi\\colon \\rquot{A^\\bul(\\Sigma)[T]}{(JT+P(T))} \\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} A^\\bul(\\Sigma')\\]\nwhich sends $T$ to $-x_\\rho$ and which verifies\n\\[ \\forall \\varrho \\in \\Sigma_1, \\qquad\n\\chi(x_\\varrho) = \\begin{cases}\nx_\\varrho+x_\\rho & \\textrm{ if $\\varrho \\prec \\sigma$}\\\\\nx_\\varrho & \\textrm{otherwise}.\n\\end{cases}\n\\]\nIn particular this gives a vector space decomposition of $A^\\bul(\\Sigma')$ as\n\\begin{align}\\label{eq:keel}\nA^\\bul(\\Sigma')\\simeq A^\\bul(\\Sigma)\\oplus A^{\\bul-1}(\\Sigma^\\sigma)T \\oplus \\dots \\oplus A^{\\bul-\\dims{\\sigma}+1}(\\Sigma^\\sigma)T^{\\dims{\\sigma}-1}.\n\\end{align}\n\\end{thm}\n\n\\begin{proof} This follows from~\\cite{Keel}*{Theorem 1 in the appendix} for the map of toric varieties $\\P_{\\Sigma'} \\to \\P_{\\Sigma}$. Here $P(T)$ is the polynomial in $A^\\bul(\\P_\\Sigma)$ whose restriction in $A^\\bul(\\Sigma^\\sigma)$ is the Chern polynomial of the normal bundle for the inclusion of toric varieties $\\P_{\\Sigma^\\sigma} \\hookrightarrow \\P_{\\Sigma}$.\n\\end{proof}\n\n\n\n\\subsection{Ascent and Descent} Situation as in Section~\\ref{sec:Keel}, let $\\Sigma$ be a unimodular fan and $\\sigma$ a cone in $\\Sigma$. Let $\\Sigma'$ be the star subdivision of $\\Sigma$ obtained by star subdividing $\\sigma$. Denote by $\\rho$ the new ray in $\\Sigma'$. Let $\\ell$ be an element of $A^1(\\Sigma)$, and denote by $\\ell^\\sigma$ the restriction of $\\ell$ to $\\Sigma^\\sigma$, i.e., the image of $\\ell$ under the restriction map $A^\\bul(\\Sigma) \\to A^\\bul(\\Sigma^\\sigma)$.\n\n\\begin{thm} \\label{thm:barycentric_subdivision} We have the following properties.\n\\begin{itemize}\n\\item \\emph{(Ascent)} Assume the property $\\HR(\\Sigma^\\sigma,\\ell^\\sigma)$ holds. Then $\\HR(\\Sigma, \\ell)$ implies $\\HR(\\Sigma',\\ell'-\\epsilon x_\\rho)$ for any small enough $\\epsilon>0$, where\n\\[ \\ell' = \\sum_{\\varrho\\in\\Sigma}\\ell(\\e_\\varrho)x_\\varrho + \\bigl(\\sum_{\\varrho \\prec \\sigma \\\\ \\dims\\varrho=1}\\ell(\\e_\\varrho)\\bigr)x_\\rho. \\]\n\n\\item \\emph{(Descent)} We have the following partial inverse: if both the properties $\\HR(\\Sigma^\\sigma,\\ell^\\sigma)$ and $\\HL(\\Sigma,\\ell)$ hold, and if we have the property $\\HR(\\Sigma',\\ell'+\\epsilon T)$ for any small enough $\\epsilon>0$, then we have $\\HR(\\Sigma, \\ell)$.\n\\end{itemize}\n\\end{thm}\n\nNotice that the definition of $\\ell'$ does not depend on the chosen representative of the class of $\\ell$ as a cone-wise linear function on $\\Sigma$.\n\n\\begin{remark} \\label{rem:keel}\nOne can check in the proof of Theorem \\ref{thm:barycentric_subdivision} below that we only use Keel's lemma, Poincar\u00e9 duality and Proposition \\ref{prop:HR}. For instance, we do not use that we are dealing with fans. Therefore, Theorem \\ref{thm:barycentric_subdivision} might be useful in other contexts where an analog of Keel's lemma holds. In particular, this remark will be useful in Section \\ref{sec:proofmaintheorem} where we need a similar ascent-descent theorem in the global case.\n\\end{remark}\n\nIn preparation for the proof, we introduce some preliminaries. Let $d$ be the dimension of $\\Sigma$. By Keel's lemma, we have $A^d(\\Sigma) \\simeq A^d(\\Sigma')$, and under this isomorphism, the canonical element $\\omega_\\Sigma$ corresponds to the canonical element $\\omega_{\\Sigma'}.$\n\nLet $\\alpha$ be a lifting of $\\omega_{\\Sigma^\\sigma}$ in $A^{d-\\dims \\sigma}(\\Sigma)$ for the restriction map $A^\\bul(\\Sigma) \\to A^\\bul(\\Sigma^{\\sigma})$. Such a lifting can be obtained for example by fixing a maximum dimensional cone $\\tau$ in $\\Sigma^{\\sigma}$; in this case we have\n\\[\\omega_{\\Sigma^\\sigma} = \\prod_{\\varrho \\prec \\tau \\\\ \\dims\\varrho=1} x_\\varrho\\]\nand a lifting $\\alpha$ of $\\omega_{\\Sigma^\\sigma}$ is given by taking the product $\\prod_{\\varrho \\prec \\~\\tau}x_\\varrho$ in $A^{d-\\dims\\sigma}(\\Sigma)$, where $\\~\\tau$ is the face of $\\Sigma$ corresponding to $\\tau$.\n\nUsing this lifting, the canonical element $\\omega_{\\Sigma'}$ can be identified with $-T^{\\dims \\sigma} \\alpha$ in $A^\\bul(\\Sigma') \\simeq \\rquot{A^\\bul(\\Sigma)[T]}{(JT+P(T))}$. Indeed, since $P(T) = 0$, we get\n\\[-T^{\\dims \\sigma} = \\sum_{j=0}^{\\dims \\sigma -1} S_{j} T^{\\dims \\sigma -j},\\]\nwith $S_j$ the $j$-th symmetric function in the variables $\\x_\\varrho$ for $\\varrho$ a ray in $\\sigma$, seen as an element of $A^\\bul(\\Sigma')$. Therefore we get\n\\begin{align*}\n-T^{\\dims \\sigma} \\alpha &= \\sum_{j=1}^{\\dims \\sigma} S_{j} \\alpha T^{\\dims \\sigma -j} \\\\\n&= \\prod_{\\varrho \\prec \\sigma\\\\ \\dims\\varrho =1} x_\\varrho \\cdot \\alpha + \\sum_{j=1}^{\\dims \\sigma -1} S_{j} \\alpha T^{\\dims \\sigma -j}.\n\\end{align*}\nSince $\\omega_{\\Sigma^{\\sigma}}=\\i^*_\\sigma(\\alpha)$ lives in the top-degree part of $A^\\bul(\\Sigma^\\sigma)$, the products $S_j \\omega_{\\Sigma^\\sigma}$ belongs to $J$. Thus the terms of the sum are all vanishing for $j=1,\\dots, \\dims \\sigma -1$. Therefore, we get\n\\[-T^{\\dims \\sigma} \\alpha = \\prod_{\\varrho \\prec \\sigma\\\\ \\dims\\varrho =1} x_\\varrho\\cdot \\alpha = \\prod_{\\varrho \\prec \\sigma\\\\ \\dims\\varrho =1} x_\\varrho \\cdot \\prod_{\\varrho \\prec \\~\\tau\\\\ \\dims\\varrho =1} x_\\varrho = \\omega_{\\Sigma} = \\omega_{\\Sigma'}.\\]\n\nWe infer that the following commutative diagram holds.\n\n\\[ \\begin{tikzcd}\nA^{d-\\dims\\sigma}(\\Sigma^\\sigma) \\arrow[r, \"\\cdot \\left(-T^{\\dims{\\sigma}}\\right)\", \"\\sim\"'] \\arrow[rd, \"\\sim\"{sloped, above}, \"\\deg\"'] & A^d(\\Sigma') \\arrow[r, \"\\sim\"'] \\arrow[d, \"\\deg\", \"\\sim\"{sloped, below}] & A^d(\\Sigma) \\arrow[ld, \"\\deg\", \"\\sim\"{sloped, above}] \\\\\n& \\corps\n\\end{tikzcd} \\]\n\nWe are now ready to give the proof of the main theorem of this section.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:barycentric_subdivision}]\n\n(Ascent)\\quad The idea for the proof of the ascent property is well-known, e.g., it is used as a way to derive Grothendieck's standard conjecture of Lefschetz type for a blow-up from the result on the base. It is also used in~\\cite{AHK}. We give the proof here. Let $\\epsilon>0$ and define the linear automorphism\n\\[S_\\epsilon\\colon A^\\bul(\\Sigma')\\to A^\\bul(\\Sigma')\\]\nof degree $0$ which is the identity map $\\id$ on $A^\\bul(\\Sigma)$ and multiplication by $\\epsilon^{-\\dims{\\sigma}\/2+k}\\id$ on each $A^\\bul(\\Sigma^\\sigma)T^k$, in the direct sum decomposition in Theorem~\\ref{thm:keel}.\n\nWe admit for the moment that $Q_{\\ell'+\\epsilon T}\\circ S_\\epsilon$ admits a limit $Q$ such that the pair $(A^\\bul(\\Sigma'),Q)$, consisting of a graded vector space and a bilinear form on this vector space, is naturally isomorphic to the pair\n\\[ \\Xi:=\\Bigl(A^\\bul(\\Sigma)\\oplus \\big(T\\cdot(\\rquot{\\mathbb{R}[T]}{T^{\\dims\\sigma-1}})\\otimes A^\\bul(\\Sigma^\\sigma)\\big), \\ell \\oplus(T\\otimes\\id+\\id\\otimes \\ell^\\sigma)\\Bigr). \\]\nIt follows from the statement~\\eqref{prop:HR:otimes_lefschetz} in Proposition~\\ref{prop:HR}, the hypothesis of the theorem, and $\\HR(\\rquot{\\mathbb{R}[T]}{T^{\\dims\\sigma-1}}, T)$, that the pair $\\Xi$ verifies $\\HR(\\Xi)$.\nUsing Keel's lemma, it follows that we have $\\HR(A^\\bul(\\Sigma'),Q)$. By Proposition~\\ref{prop:HRbis}, the property $\\HR(A^\\bul(\\Sigma'),Q_{\\ell'+\\epsilon T}\\circ S_\\epsilon)$ holds for any small enough value of $\\epsilon>0$. Finally, applying Proposition~\\ref{prop:HR}, we get $\\HR(A^\\bul(\\Sigma'),Q_{\\ell'+\\epsilon T})$, which finishes the proof of the ascent part of our theorem.\n\n\\medskip\n\nBefore turning to the proof of the existence of the limit, let us explain the proof of the descent part.\n\n\\bigskip\n\n\\noindent(Descent)\n\n\\smallskip\nWe again use Proposition~\\ref{prop:HR}. By our assumption, we have the properties $\\HL(A^\\bul(\\Sigma), \\ell)$ and $\\HR(A^\\bul(\\Sigma^\\sigma,\\ell^\\sigma))$, from which we get the property\n\\[ \\HR\\Bigl((T\\cdot(\\rquot{\\mathbb{R}[T]}{T^{\\dims\\sigma-1}})\\otimes A^\\bul(\\Sigma^\\sigma)\\big), (T\\otimes\\id+\\id\\otimes \\ell^\\sigma)\\Bigr). \\]\n\nDefining $\\Xi$ as in the first part of the proof, we deduce $\\HL(\\Xi)$ by applying Proposition~\\ref{prop:HR}. By the statement~\\eqref{prop:HR:oplus} in that proposition, it will be enough to prove\n$\\HR(\\Xi)$ in order to get $\\HR(A^\\bul(\\Sigma), \\ell)$.\n\nNote that by the hypothesis in the theorem, we have $\\HR(A^\\bul(\\Sigma'),Q_{\\ell'+\\epsilon T})$ for small enough values of $\\epsilon>0$. Using Proposition~\\ref{prop:HR}, we deduce that $\\HR(A^\\bul(\\Sigma'),Q_{\\ell'+\\epsilon T}\\circ S_\\epsilon)$ holds for $\\epsilon>0$ small enough. It follows that we have as well $\\HL(A^\\bul(\\Sigma'),Q_{\\ell'+\\epsilon T}\\circ S_\\epsilon)$.\n\nApplying now Proposition~\\ref{prop:HRbis}, since $\\HR$ is a closed condition restricted to the space of bilinear forms which verify $\\HL$, we deduce that $\\HR(\\Xi)$ holds: indeed, $Q$\nis the limit of $Q_{\\ell'+\\epsilon T}\\circ S_\\epsilon$ and $\\HR(A^\\bul(\\Sigma'),Q_{\\ell'+\\epsilon T}\\circ S_\\epsilon)$ and $\\HL(A^\\bul(\\Sigma'),Q_{\\ell'+\\epsilon T}\\circ S_\\epsilon)$ hold for $\\epsilon >0 $ small enough.\n\n\\vspace{.7cm}\n\nWe are thus left to prove the last point, namely the existence of the limit $\\lim_{\\epsilon \\to 0}\\, Q_{\\ell'+\\epsilon T}\\circ S_\\epsilon$.\n\nConsider the decomposition~\\eqref{eq:keel} given in Keel's lemma. The decomposition for degree $k$ part $A^{k}(\\Sigma')$ of the Chow ring has pieces $A^{k}(\\Sigma)$ and $A^{k-i}(\\Sigma^\\sigma) T^{i}$ for $1 \\leq i \\leq \\min\\{k, \\dims \\sigma -1\\}$. Consider the bilinear form $\\Phi_{A^\\bul(\\Sigma')}$ given by the degree map of $A^\\bul(\\Sigma')$.\n\nFor this degree bilinear form, each piece $A^{k-i}(\\Sigma^\\sigma)T^i$ is orthogonal to the piece $A^{d-k}(\\Sigma)$ as well as to the piece $A^{d-k-j}(\\Sigma^\\sigma)T^j$ for $j < \\dims \\sigma -i$, in the decomposition of $A^{d-k}(\\Sigma')$ given in Keel's lemma.\n\nWe now work out the form of the matrix of the bilinear forms $Q_{\\ell'+\\epsilon T}\\circ S_\\epsilon$ in degree $k$ with respect to the decomposition given in \\eqref{eq:keel}.\n\nFirst, note that for $a,b\\in A^k(\\Sigma)$, we have\n\\[ Q_{\\ell'+\\epsilon T}(a,b)=\\deg(a\\ell^{\\prime d-2k}b)+\\O(\\epsilon^{\\dims \\sigma}). \\]\n\nSecond, for an element $a\\in A^k(\\Sigma)$ and an element $b\\in A^{k-i}(\\Sigma^\\sigma)T^i$, we get the existence of an element $c\\in A^{d-2k-\\dims\\sigma+i}(\\Sigma')$ such that\n\\[ Q_{\\ell'+\\epsilon T}(a,b)=\\deg(ac(\\epsilon T)^{\\dims\\sigma-i}\\epsilon^{-\\dims\\sigma\/2+i}b)=\\O(\\epsilon^{\\dims \\sigma\/2}). \\]\n\nThird, for an element $a\\in A^{k-i}(\\Sigma^\\sigma)T^i$ and an element $b\\in A^{k-j}(\\Sigma^\\sigma)T^j$, with $i+j\\leq\\dims\\sigma$, we get\n\\begin{align*}\nQ_{\\ell'+\\epsilon T}(a,b) &= \\deg\\Big(\\epsilon^{-\\dims\\sigma\/2+i}a\\big(C(\\epsilon T)^{\\dims\\sigma-i-j}\\ell'^{d-2k-\\dims\\sigma+i+j}+\\O(\\epsilon^{\\dims\\sigma-i-j+1})\\big)\\epsilon^{-\\dims\\sigma\/2+j}b\\Big) \\\\\n  &=C\\deg_{A^\\bul(\\Sigma^\\sigma)}(a(\\ell^\\sigma)^{d-\\dims\\sigma-2k+i+j}b)+\\O(\\epsilon),\n\\end{align*}\nwith $C=\\binom{d-2k}{\\dims\\sigma-i-j}$.\n\nFinally, if $a\\in A^{k-i}(\\Sigma^\\sigma)T^i$ and $b\\in A^{k-j}(\\Sigma^\\sigma)T^j$, with $i+j>\\dims\\sigma$, we get\n\\[ Q_{\\ell'+\\epsilon T}(a,b)=\\deg(\\epsilon^{-\\dims\\sigma\/2+i}a\\O(1)\\epsilon^{-\\dims\\sigma\/2+j}b)=\\O(\\epsilon^{i+j-\\dims\\sigma}). \\]\n\n\\medskip\n\nDoing the same calculation for the bilinear form $Q_{\\~L}$ associated to the linear map of degree one $\\~L\\colon A^\\bul(\\Sigma')\\to A^{\\bul+1}(\\Sigma')$ given by the matrix\n\\[ \\begin{pmatrix}\n\\ell \\\\\n0 &\\ell^\\sigma & & \\makebox(0,0){\\huge0} \\\\\n & T & \\ell^\\sigma \\\\[1ex]\n & \\makebox(0,0){\\huge0} & \\ddots & \\ddots \\\\[1ex]\n & & & T & \\ell^\\sigma\n\\end{pmatrix} \\]\nrelative to the decomposition~\\eqref{eq:keel} given in Keel's lemma, we see that\n\\[\\lim_{\\epsilon \\to 0}\\,Q_{\\ell'+\\epsilon T} = Q_{\\~L}. \\]\nTo conclude, we observe that the pair $(A^\\bul(\\Sigma'), Q_{\\~L})$ is isomorphic to the previously introduced pair,\n\\[ \\Xi=\\Bigl(A^\\bul(\\Sigma)\\oplus \\big(T\\cdot(\\rquot{\\mathbb{R}[T]}{T^{\\dims\\sigma-1}})\\otimes A^\\bul(\\Sigma^\\sigma)\\big), \\ell\\oplus(T\\otimes\\id+\\id\\otimes \\ell^\\sigma)\\Bigr), \\]\nas claimed.\n\\end{proof}\n\n\n\n\\subsection{Weak factorization theorem} Two unimodular fans with the same support are called \\emph{elementary equivalent} if one can be obtained from the other by a barycentric star subdivision. The \\emph{weak equivalence} between unimodular fans with the same support is then defined as the transitive closure of the elementary equivalence relation.\n\n\\begin{thm}[Weak factorization theorem] \\label{thm:equivalent_fan}\nTwo unimodular fans with the same support are always weakly equivalent.\n\\end{thm}\n\\begin{proof}\nThis is Theorem A of~\\cite{Wlo97} proved independently by Morelli~\\cite{Mor96} and expanded by Abramovich-Matsuki-Rashid, see~\\cite{AMR}.\n\\end{proof}\n\nWe will be interested in the class of quasi-projective unimodular fans on the same support. In this class, we define the \\emph{weak equivalence} between quasi-projective unimodular fans as the transitive closure of the elementary equivalence relation between quasi-projective unimodular fans with the same support. This means all the intermediate fans in the star subdivision and star assembly are required to remain quasi-projective.\n\n\\begin{thm}[Weak factorization theorem for quasi-projective fans] \\label{thm:equivalent_fan2}\nTwo quasi-projective unimodular fans with the same support are weakly equivalent within the class of quasi-projective fans.\n\\end{thm}\n\\begin{proof}\nThis can be obtained from relevant parts of~\\cites{Wlo97, Mor96, AKMW} as  discussed and generalized by Abramovich and Temkin in~\\cite{AT}*{Section 3}.\n\\end{proof}\n\nWe thank Kenji Matsuki and Kalle Karu for helpful correspondence and clarification on this subject.\n\n\n\n\\subsection{Ample classes}\n\nLet $\\Sigma\\subseteq N_\\mathbb{R}$ be a unimodular Bergman fan. Let $f$ be a strictly convex cone-wise linear function on $\\Sigma$. Recall that if $\\varrho$ is a ray in $\\Sigma_1$, we denote by $\\e_\\varrho$ the primitive element of $\\varrho$ and by $x_\\varrho$ the corresponding element of $A^1(\\Sigma)$. We denote by $\\ell(f)$ the element of $A^1(\\Sigma)$ defined by\n\\[ \\ell(f):=\\sum_{\\varrho\\in\\Sigma^1}f(\\e_\\varrho)x_\\varrho. \\]\n\n\\begin{prop} \\label{prop:ell_f_independent}\nNotations as above, let $\\sigma$ be a cone of $\\Sigma$ and let $\\phi$ be a linear form on $N_\\mathbb{R}$ which coincides with $f$ on $\\sigma$. The function $f-\\phi$ induces a cone-wise linear function on $\\Sigma^\\sigma$ which we denote by $f^\\sigma$. Then\n\\[ \\ell(f^\\sigma)=\\i^*_{\\conezero\\prec\\sigma}(\\ell(f)). \\]\nIn particular, $\\ell(f^\\sigma)$ does not depend on the chosen $\\phi$.\n\nMoreover, if $f$ is strictly convex on $\\Sigma$, then $f^\\sigma$ is strictly convex on $\\Sigma^\\sigma$.\n\\end{prop}\n\n\\begin{proof}\nWe write $\\varrho\\sim\\sigma$ if $\\varrho$ is a ray in the link of $\\sigma$, i.e., $\\varrho\\not\\in\\sigma$ and $\\sigma+\\varrho$ is a face of $\\Sigma$. Such rays are in one-to-one correspondence with rays of $\\Sigma^\\sigma$. Recall the following facts:\n\\[ \\ell(\\phi)=0, \\qquad \\i^*_{\\conezero\\prec\\sigma}(x_\\varrho)=0 \\text{ if $\\varrho\\not\\sim\\sigma$ and $\\varrho\\not\\in\\sigma$}, \\qquad f(\\e_\\varrho)=\\phi(\\e_\\varrho) \\text{ if $\\varrho\\prec\\sigma$}. \\]\nThen we have\n\\begin{align*}\n\\i^*_{\\conezero\\prec\\sigma}(\\ell(f))\n  &= \\i^*_{\\conezero\\prec\\sigma}(\\ell(f-\\phi)) \\\\\n  &= \\sum_{\\varrho\\in\\Sigma_1}(f-\\phi)(\\e_\\varrho)\\i^*_{\\conezero\\prec\\sigma}(x_\\varrho) = \\sum_{\\varrho\\in\\Sigma_1 \\\\ \\varrho\\sim\\sigma}(f-\\phi)(\\e_\\varrho)\\i^*_{\\conezero\\prec\\sigma}(x_\\varrho) \\\\\n  &= \\sum_{\\varrho\\in\\Sigma^\\sigma\\\\ \\dims\\varrho=1}f^\\sigma(\\e_{\\varrho})x_{\\varrho} = \\ell(f^\\sigma). \\qedhere\n\\end{align*}\n\nFor the last point, let $f$ be a strictly convex cone-wise linear function on $\\Sigma$. We have to prove that $f^\\sigma$ is strictly convex. Let $\\zeta$ be a cone of $\\Sigma^\\sigma$. The cone $\\zeta$ corresponds to a unique cone $\\~\\zeta$ in $\\Sigma$ that contains $\\sigma$. Since $f$ is strictly convex, there exists a linear form $\\psi$ on $N_\\mathbb{R}$ such that $f-\\psi$ is strictly positive around $\\~\\zeta$ (in the sense of Section \\ref{subsec:convexite}). Then $\\psi-\\phi$ is zero on $\\sigma$, thus induces a linear form $\\psi^\\sigma$ on $N^\\sigma_\\mathbb{R}$. We get that $f^\\sigma-\\psi^\\sigma$ is strictly positive around $\\zeta$, thus $f^\\sigma$ is strictly convex around $\\zeta$. We infer that $f^\\sigma$ is strictly convex on $\\Sigma^\\sigma$.\n\\end{proof}\n\n\\begin{defi}\nLet $\\Sigma$ be a Bergman fan. An element $\\ell\\in A^1(\\Sigma)$ is called an \\emph{ample class} if there exists a strictly convex cone-wise linear function $f$ on $\\Sigma$ such that $\\ell=\\ell(f)$.\n\\end{defi}\n\n\n\n\\subsection{Main Theorem}\n\nHere is the main theorem of this section.\n\n\\begin{thm}\\label{thm:mainlocal} Any quasi-projective unimodular Bergman fan $\\Sigma$ verifies $\\HR(\\Sigma, \\ell)$ for any ample class $\\ell \\in A^1(\\Sigma)$.\n\\end{thm}\n\nThe following proposition is the base of our proof. It allows to start with a fan on any Bergman support which verifies the Hodge-Riemann relations, and then use the ascent and descent properties to propagate the property to any other fan on the same support.\n\n\\begin{prop}\\label{prop:baseHR}\nFor any Bergman support, there exists a quasi-projective unimodular fan $\\Sigma$ on this support and a strictly convex element $\\ell \\in A^1(\\Sigma)$ which verifies $\\HR(\\Sigma, \\ell)$.\n\\end{prop}\n\nWe prove both statements, in the theorem and proposition, by induction on the dimension of the ambient space, i.e., on the size of the ground set of the matroid underlying the support of the Bergman fan.\nSo suppose by induction that the theorem and the proposition hold for all Bergman support of $\\mathbb{R}^k$ with $k<n$, and for any quasi-projective unimodular fan with a Bergman support lying in $\\mathbb{R}^k$ for $k<n.$\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:baseHR}]\nLet $\\Ma$ be a simple matroid on the ground set $\\{0,\\dots,n\\}$.\n\nIf $\\Ma$ is the free matroid, then we have $\\supp{\\Sigma_\\Ma}=\\mathbb{R}^n$. In this case, we can take any projective unimodular fan. For example, the fan of the projective space $\\P^n$, equivalently of $\\TP^{n}$, $\\Sigma_{\\TP^n}$ has Chow ring isomorphic to $\\rquot{\\mathbb{R}[\\x]}{\\x^{n+1}}$. In this case, we can take $\\ell = \\x$, and we have $\\HR(\\rquot{\\mathbb{R}[\\x]}{\\x^{n+1}},\\x)$.\n\n\\medskip\n\nOtherwise, $\\Ma$ is not the free matroid. Let $i$ be a proper element of $\\Ma$. By the induction hypothesis, there exists a quasi-projective unimodular fan $\\Sigma'$ of support $\\supp{\\Sigma_{\\Ma\\del{i}}}$. Doing some blow-ups if necessary, we can assume that $\\Sigma'$ contains a subdivision $\\Delta'$ of $\\Sigma_{\\Ma\\contr{i}}$ (cf. Section \\ref{sec:triangulation} for more details on this argument). Let $\\ell'$ be a strictly convex function on $\\Sigma'$. Modifying $\\ell'$ by adding a linear function if necessary, we can suppose without loss of generality that $\\ell'$ is strictly positive on $\\supp{\\Sigma'}\\setminus\\{0\\}$. Let $\\Sigma = \\modtrop{\\Delta'}{\\Sigma'}$, the unimodular fan with support $\\supp{\\Sigma_\\Ma}$ obtained as the tropical modification of $\\Delta'$ in $\\Sigma'$. Using Proposition~\\ref{prop:modification}, we deduce $\\HR(\\Sigma, \\ell)$ from $\\HR(\\Sigma_{\\Ma\\del i},\\ell')$ where $\\ell$ is the cone-wise linear function on $\\Sigma$ which takes value $0$ on the new ray $\\varrho_i$ of $\\Sigma$ and which coincides on other rays with $\\Gamma_*(\\ell')$ where $\\Gamma\\colon \\Sigma'\\hookrightarrow\\Sigma$ is the natural injective map. One easily verifies that $\\ell$ is strictly convex. So the proposition follows.\n\\end{proof}\n\n\\begin{prop} \\label{prop:HR-oneall}\nFor any unimodular fan $\\Sigma$ of dimension $d$, we have the equivalence of the following statements.\n\\begin{itemize}\n\\item $\\HR(\\Sigma, \\ell)$ is true for any strictly convex element $\\ell \\in A^1(\\Sigma)$.\n\\item $\\HR(\\Sigma, \\ell)$ is true for a strictly convex element $\\ell \\in A^1(\\Sigma)$.\n\\end{itemize}\n\\end{prop}\n\\begin{proof} Let $\\ell$ be a strictly convex piecewise linear function on $\\Sigma$. For all $\\sigma\\in \\Sigma$, we get a strictly convex function $\\ell^\\sigma$ on $\\Sigma^\\sigma$. By the hypothesis of our induction, we know that $\\HR(\\Sigma^\\sigma, \\ell^\\sigma)$ holds. By Proposition~\\ref{prop:local_HR} we thus get $\\HL(\\Sigma, \\ell)$.\n\nBy Proposition \\ref{prop:HRbis} we know that $\\HR(\\Sigma,\\ell)$ is an open and closed condition on the set of all $\\ell$ which satisfy $\\HL(\\Sigma,\\ell)$. In particular, if there exists $\\ell_0$ in the cone of strictly convex elements which verifies $\\HR(\\Sigma,\\ell_0)$, any $\\ell$ in this cone should verify $\\HR(\\Sigma,\\ell)$.\n\\end{proof}\n\nLet $\\Sigma$ be a unimodular fan of dimension $d$, $\\ell$ a strictly convex element on $\\Sigma$, and let $\\Sigma'$ be the fan obtained from $\\Sigma$ by star subdividing a cone $\\sigma\\in\\Sigma$. Denote by $\\rho$ the new ray in $\\Sigma'$, and let\n\\[\\ell':=\\ell+\\bigl(\\sum_{\\varrho \\prec \\sigma \\\\ \\dims\\varrho=1}\\ell(\\e_\\varrho)\\bigr)x_\\rho.\\]\n\nThe following is straightforward.\n\n\\begin{prop} For any small enough $\\epsilon>0$, the element $\\ell' - \\epsilon x_\\rho$ of $A^1(\\Sigma')$ is strictly convex.\n\\end{prop}\n\n\\begin{prop} \\label{prop:HR-trans}\nNotations as above, the following statements are equivalent.\n\\begin{enumerate}[label={\\textnormal{(\\textit{\\roman*})}}]\n\\item \\label{hr:trans1} We have $\\HR(\\Sigma, \\ell)$.\n\\item \\label{hr:trans2} The property $\\HR(\\Sigma',\\ell'-\\epsilon x_\\rho)$ holds for any small enough $\\epsilon>0$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nIt will be enough to apply Theorem~\\ref{thm:barycentric_subdivision}. A direct application of the theorem, and the fact that $T$ is identified with $-x_\\rho$ via Keel's lemma, leads to the implication $\\ref{hr:trans1} \\Rightarrow \\ref{hr:trans2}$.\n\nWe now explain the implication $\\ref{hr:trans2} \\Rightarrow \\ref{hr:trans1}$. By the hypothesis of our induction, we have $\\HR(\\Sigma^\\sigma, \\ell^\\sigma)$. Moreover, Proposition \\ref{prop:local_HR} and our induction hypothesis imply that we have $\\HL(\\Sigma, \\ell)$.\nApplying now the descent part of Theorem~\\ref{thm:barycentric_subdivision} gives the result.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:mainlocal}] Let $\\Ma$ be a matroid on the ground set $\\{0,\\dots, n\\}$. By Proposition~\\ref{prop:baseHR} there exists a quasi-projective unimodular fan $\\Sigma_0$ with support $\\supp{\\Sigma_\\Ma}$ and a strictly convex element $\\ell_0 \\in A^1(\\Sigma_0)$ such that $\\HR(\\Sigma_0, \\ell_0)$ holds. By Propositions~\\ref{prop:HR-trans} and~\\ref{prop:HR-oneall}, for any quasi-projective fan $\\Sigma$ in the quasi-projective weak equivalence class of $\\Sigma_0$, so with support $\\supp{\\Sigma_\\Ma}$, and any strictly convex element $\\ell \\in A^1(\\Sigma)$, we get $\\HR(\\Sigma, \\ell)$. The theorem now follows in dimension $n$ by applying Theorem~\\ref{thm:equivalent_fan2}\n\\end{proof}\n\n\n\n\\subsection{Examples} We finish this section with some concrete examples of fans verifying the Hodge-Riemann relations with respect to ample and non-ample classes.\\label{ex:convex_U33}\n\n\\medskip\n\n\n\\subsubsection{} Let $\\Sigma$ be the complete Bergman fan associated to the uniform matroid $U_{3,3}$ on the ground set $\\{0,1,2\\}$ as illustrated in Figure \\ref{fig:Bergman_fan_U33}.\n\\begin{figure}[ht]\n\\caption{The Bergman fan of $U_{3,3}$} \\label{fig:Bergman_fan_U33}\n\\begin{tikzpicture}\n\\clip (0, 0) circle (2);\n\\fill[color=gray!10] (-2,-2) rectangle (2,2);\n\\draw[color=gray!25, scale=.6] (-4,-4) grid (4,4);\n\\draw  (0 , 0)\n  edge[\"$\\rho_1$\", near end]     (2 , 0)\n  edge[\"$\\rho_{12}$\"{above left=-4pt}]  (2 , 2)\n  edge[\"$\\rho_{2}$\", near end]   (0 , 2)\n  edge[\"$\\rho_{02}$\", near end]  (-2, 0)\n  edge[\"$\\rho_{0}$\"{below right=-4pt}]   (-2,-2)\n  edge[\"$\\rho_{01}$\", near end]  (0 ,-2);\n\\end{tikzpicture}\n\\end{figure}\nThe flats of $U_{3,3}$ are $\\Cl(U_{3,3}):=\\{1,12,2,02,0,01\\}$ (where $ij$ denote the set $\\{i,j\\}$). The rays of $\\Sigma$ correspond to the flats of $U_{3,3}$ and are denoted $\\rho_F$ with $F\\in\\Cl(U_{3,3})$. These rays corresponds to elements of $A^1(\\Sigma)$ denoted $x_F$, for $F\\in\\Cl(U_{3,3})$. We have\n\\[ \\dim(A^1(\\Sigma))=\\card{\\Cl(U_{3,3})}-\\dim\\bigl(\\mathbb{R}^{2^\\dual}\\bigr)=4. \\]\nWe have a natural degree map $\\deg\\colon A^2(\\Sigma)\\to\\mathbb{R}$ which verifies the following.\n\\[ \\deg(x_Fx_G)=\\begin{cases}\n  -1 & \\text{if $F=G$,} \\\\\n  1  & \\text{if $F\\neq G$ and $F$ and $G$ are comparable,} \\\\\n  0  & \\text{if $F$ and $G$ are not comparable.}\n\\end{cases} \\]\n\n\\medskip\n\nLet $\\ell:=x_{1}+x_{12}+x_{2}+x_{02}+x_{0}+x_{01}\\in A^1(\\Sigma)$. This element corresponds to a strictly convex cone-wise linear function on $\\Sigma$, thus, it must verify the Hodge-Riemann relations $\\HR(\\Sigma,\\ell)$. Let us check that this is the case. We have to check that $\\deg(\\ell^2)>0$ and that\n\\[ \\begin{array}{rccc}\nQ_1\\colon & A^1(\\Sigma)  \\times  A^1(\\Sigma) & \\to     & \\mathbb{R},       \\\\\n          &( x            ,       y )        & \\mapsto & \\deg(xy),\n\\end{array} \\]\nhas signature $-2=2\\dim(A^0)-\\dim(A^1)$. We have\n\\[ \\deg(\\ell x_1)=\\deg(x_1^2+x_1x_{12}+x_1x_{01})=1, \\]\nand, by symmetry, $\\deg(\\ell x_F)=1$ for every $F\\in\\Cl(U_{3,3})$. Thus, $\\deg(\\ell^2)=6$.\n\nFor $Q_1$ we have an orthogonal basis $(x_0, x_1, x_2, x_0+x_{01}+x_1)$. Moreover $\\deg(x_0^2)=\\deg(x_1^2)=\\deg(x_2^2)=-1$ and\n\\[ \\deg((x_0+x_{01}+x_1)^2)=\\deg(x_0^2+x_{01}^2+x_1^2+2x_0x_{01}+2x_{01}x_1)=1. \\]\nThus, the signature of $Q_1$ is $-2$ and the Hodge-Riemann relations are verified.\n\n\\medskip\n\n\n\\subsubsection{} Let us give another example in the same Chow ring treated in the previous paragraph. Let $\\ell'=x_1+x_{12}+x_2$. The cone-wise linear function associated to $\\ell'$ is not convex. It is even zero around $\\rho_0$, which implies that $\\i^*_{\\conezero\\ssubface\\rho_0}(\\ell')=0$. Nevertheless, $\\ell'$ verifies the Hodge-Riemann relations $\\HR(\\Sigma, \\ell')$. We only have to check that $\\deg(\\ell'^2)>0$, since $Q_1$ does not depend on $\\ell'$ and we have already checked that $Q_1$ has signature $-2$. We have\n\\[ \\deg((x_1+x_{12}+x_2)^2)=\\deg(x_1^2+x_{12}^2+x_2^2+2x_1x_{12}+2x_{12}x_2)=1>0. \\]\nThis gives us an example of a non-ample element verifying the Hodge-Riemann relations.\n\n\\medskip\n\n\n\\subsubsection{} As a last example, let $\\Sigma$ be the Bergman fan of the uniform matroid $U_{3,4}$ on the ground set $\\{0,1,2,3\\}$, and let $\\ell=(2+2\\epsilon)x_0+x_2+x_3+\\epsilon x_{01}+(3+\\epsilon)(x_{02}+x_{03})-x_{23}\\in A^1(\\Sigma)$ where $\\epsilon$ is a small positive number. Then $\\ell$ is not ample, but we have $\\HR(\\Sigma, \\ell)$. Indeed, we have something stronger: $\\ell^\\sigma$ is ample for any $\\sigma\\in\\Sigma\\setminus\\{\\conezero\\}$ where $\\ell^\\sigma = \\i^*_{\\conezero \\prec \\sigma}(\\ell)$.\n\n\n\n\n\n\n\\section{Quasi-projective unimodular triangulations of polyhedral complexes}\\label{sec:triangulation}\n\nIn this section we show that any rational polyhedral complex in $\\mathbb{R}^n$ admits a simplicial subdivision which is both \\emph{quasi-projective} and \\emph{unimodular} with respect to the lattice $\\frac 1k \\mathbb{Z}^n$, for some natural number $k$, in a sense which we will precise in a moment. Moreover, we will show this property holds for any \\emph{tropical compactification} of $X$. The novelty here is in ensuring the \\emph{convexity property} underlying the definition of quasi-projective triangulations which will be the crucial property used in what will follow later, as well as in ensuring the unimodularity of unbounded polyhedra which form the polyhedral subdivision of $X$. The theorem thus extends the pioneering result proved in~\\cite{KKMS}, in Mumford's proof of the semistable reduction theorem, as well as previous variants proved in~\\cite{IKMZ} and~\\cite{Wlo97}.\n\n\n\n\\subsection{Statement of the triangulation theorem.}\n\nThis theorem corresponds to Theorem \\ref{thm:regulartriangulations} stated in the introduction. However, we add a technical result about the existence of strictly convex functions to the statement which  will be needed later.\n\n\\begin{thm}[Triangulation theorem] \\label{thm:triangulation_unimodulaire_convexe}\nLet $X$ be a rational polyhedral complex in $\\mathbb{R}^n$. There exists an integer $k$ and a triangulation $X'$ of $X$ which is quasi-projective and unimodular with respect to the lattice $\\frac 1k \\mathbb{Z}^n$.\n\nMoreover we ensure that $X'_\\infty$ is a fan, and that there exists a strictly convex function $f$ on $X'$ such that $f_\\infty$ is well-defined and strictly convex.\n\\end{thm}\nThe terminology will be introduced in the next section.\n\n\\medskip\n\nFurthermore, we will prove the following theorem which will be later used in Section \\ref{sec:projective_bundle_theorem}.\n\n\\begin{thm}[Unimodular triangulation preserving the recession fan] \\label{thm:unimodular_preserving_recession}\nLet $X$ be a polyhedral complex in $\\mathbb{R}^n$ such that its pseudo-recession fan $X_\\infty$ is a unimodular fan. Then there exists a positive integer $k$ and a subdivision $Y$ of $X$ which is unimodular with respect to $\\frac1k\\mathbb{Z}^n$ and which verifies $Y_\\infty=X_\\infty$.\n\\end{thm}\n\n\n\n\\subsection{Preliminary definitions and constructions}\n\nThis section presents certain constructions which will be needed later. While definitions are given here for the case of polyhedral complexes and fans, we note that the majority of the constructions naturally extend to the case of polyhedral pseudo-complexes and pseudo-fans.\n\n\n\\subsubsection{Restriction, intersection, and hyperplane cut}\n\nLet $X$ be a polyhedral complex of $\\mathbb{R}^n$ and let $S$ be a subset of $\\mathbb{R}^n$. The \\emph{restriction of $X$ to $S$} denoted by $X\\rest S$ is by definition the polyhedral complex consisting of all the faces of $X$ which are included in $S$. If $S$ is an affine hyperplane, an affine half-space, or more generally a polyhedron of $\\mathbb{R}^n$, the \\emph{intersection of $X$ with $S$} denoted by $X\\cap S$ is the polyhedral complex whose faces are $\\delta \\cap S$ for any face $\\delta$ of $X$, i.e.,\n\\[ X\\cap S:=\\bigl\\{\\delta\\cap S\\st \\delta\\in X\\bigr\\}. \\]\nFor an affine hyperplane $H$ in $\\mathbb{R}^n$, the \\emph{hyperplane cut of $X$ by $H$} denoted by $X\\cdot H$ is the subdivision of $X$ defined by\n\\[ X\\cdot H:=(X\\cap H)\\cup(X\\cap H_+)\\cup(X\\cap H_-), \\]\nwhere $H_+$ et $H_-$ are the two half-spaces of $\\mathbb{R}^n$ defined by $H$.\n\n\n\\subsubsection{Slicing with respect to a pencil of hyperplanes} Let $\\Sigma$ be a fan of $\\mathbb{R}^n$. For any cone $\\sigma \\in \\Sigma$, choose a collection of half-spaces $H^+_{\\sigma,1}, \\dots, H^+_{\\sigma,k_\\sigma}$ for $k_\\sigma \\in \\mathbb{Z}_{>0}$ each defined by hyperplanes $H_{\\sigma,1}, \\dots, H_{\\sigma,k_\\sigma}$, respectively, such that $\\sigma$ is the intersection of these half-spaces.\nWe call the collection of hyperplanes\n\\[\\pen_\\Sigma:= \\Bigl\\{H_{\\sigma, j} \\, \\st \\, \\sigma \\in \\Sigma \\quad \\textrm{and} \\quad j =1, \\dots, k_\\sigma\\Bigr\\}\\]\n\\emph{a pencil of hyperplanes defined by $\\Sigma$}.\n\n\\medskip\n\nLet now $\\Delta$ be a complete fan in $\\mathbb{R}^n$ and let $\\Sigma$ be a second (arbitrary) fan in $\\mathbb{R}^n$. The \\emph{slicing of $\\Delta$ with a pencil of hyperplanes $\\pen_\\Sigma$} defined by $\\Sigma$ is the fan denoted by $\\Delta \\cdot \\pen_\\Sigma$ and defined by\n\\[ \\Delta\\cdot \\pen_{\\Sigma} := \\Delta \\cdot H_{\\sigma_1,1} \\cdot H_{\\sigma_1,2}\\cdots H_{\\sigma_1,k_{\\sigma_1}}\\cdots H_{\\sigma_m,k_{\\sigma_m}}, \\]\ni.e., defined by starting from $\\Delta$ and successively taking hyperplane cuts by elements $H_{\\sigma,j}$ of the pencil $ \\pen_\\Sigma$.\n\n\\begin{prop} \\label{prop:decoupe_subdivision}\nThe slicing of a complete fan with a pencil of hyperplanes $\\pen_\\Sigma$ defined by a fan $\\Sigma$ necessarily contains a subdivision of $\\Sigma$.\n\\end{prop}\n\n\\begin{proof} First note that if $\\Delta$ is a fan in $\\mathbb{R}^n$ and $H$ is a hyperplane, all the cones $\\delta \\in \\Delta \\cdot H$ live in one of the two half-spaces $H^+$ and $H^-$ defined by $H$. This property holds as well for any subdivision of $\\Delta\\cdot H$.\n\nLet now $\\Delta$ be complete. Let $\\pen_\\Sigma$ be a pencil of hyperplanes defined by the fan $\\Sigma$ consisting of hyperplanes $H_{\\sigma, j}$ for $\\sigma \\in \\Sigma$ and $j=1, \\dots, k_\\sigma$ as above. Let $\\Delta' = \\Delta \\cdot \\pen_\\Sigma$. In order to prove the proposition, we will need to show the following two properties:\n\\begin{enumerate}\n\\item \\label{enum:pencil:1} the support of the restriction $\\Delta'\\rest{\\supp{\\Sigma}}$ is the entire support $\\supp{\\Sigma}$; and\n\\item \\label{enum:pencil:2} any face of $\\Delta'\\rest{\\supp{\\Sigma}}$ is included in a face of $\\Sigma$.\n\\end{enumerate}\n\nIn order to prove \\ref{enum:pencil:1}, let $x$ be a point in $\\supp{\\Sigma}$. It will be enough to show the existence of a face of $\\Delta'$ which contains $x$ and which is entirely in $\\supp{\\Sigma}$.\n\nLet $\\delta$ be the smallest face of $\\Delta'$ which contains $x$; $\\delta$ is the face which contains $x$ in its interior. Let $\\sigma \\in \\Sigma$ be a face which contains $x$. We will show that $\\delta \\subseteq \\sigma$.\n\nBy what preceded, for each $i \\in \\{1, \\dots, k_\\sigma\\}$, there exists $\\epsilon_i \\in \\{+, -\\} $ such that $\\delta$ is included in $H^{\\epsilon_i}_{\\sigma,i}$. Recall that by definition, we have $\\sigma=\\cap_{i=1}^{k_\\sigma}H^+_{\\sigma,i}$. We will show that we can take $\\epsilon_i = +$ for each $i$, which gives the result. Suppose $\\epsilon_i=-$ for an $i\\in \\{1, \\dots, k_\\sigma\\}$. In particular, $x$ belongs to both half-spaces $H^-_{\\sigma,i}$ and $H^+_{\\sigma,i}$, and so $x\\in H_{\\sigma,i}$. Since $\\delta\\cap H_{\\sigma,i}$ is a non-empty face of $\\delta$ which contains $x$, by minimality of $\\delta$, we infer that $\\delta\\subset H_{\\sigma,i}$ which shows that we can choose $\\epsilon_i = +$. This proves \\ref{enum:pencil:1}.\n\nIn order to prove the second assertion, let $\\delta$ be a face of the restriction $\\Delta'\\rest{\\supp{\\Sigma}}$. Let $x$ be a point in the interior of $\\delta$, so $\\delta$ is the smallest face of $\\Delta'$ which contains $x$. Let $\\sigma$ be a face of $\\Sigma$ which contains $x$. As we showed in the proof of \\ref{enum:pencil:1}, we have the inclusion $\\delta \\subseteq \\sigma$, which is the desired property in \\ref{enum:pencil:2}.\n\nIt follows that $\\Delta'\\rest{\\supp{\\Sigma}}$ is a subdivision of $\\Sigma$ and the proposition follows.\n\\end{proof}\n\n\n\\subsubsection{Blow-ups and existence of triangulations} Let $\\sigma$ be a cone in $\\mathbb{R}^n$ and let $x$ be a vector of $\\mathbb{R}^n \\setminus \\{0\\}$. The \\emph{blow-up of $\\sigma$ at $x$} is the fan denoted by $\\sigma_{(x)}$ and defined by\n\\[ \\sigma_{(x)}:=\\begin{cases}\n\\displaystyle \\bigcup_{\\substack{\\tau\\prec\\sigma \\\\ x\\not\\in\\tau}} \\bigl\\{\\,\\tau,\\tau+\\mathbb{R}_+x\\, \\bigr\\}& \\text{if $x \\in\\sigma$,} \\\\[2em]\n\\face{\\sigma} & \\text{otherwise.}\n\\end{cases} \\]\nNote that $\\sigma_{(x)}$ is a polyhedral subdivision of $\\face{\\sigma}$. Moreover, the definition depends only on the half line $\\mathbb{R}_+x$.\n\nMore generally, if $\\Sigma$ is a fan, we define the \\emph{blow-up of $\\Sigma$ at $x$} to be the fan denoted by $\\Sigma_{(x)}$ and defined as the union of blow-ups $\\sigma_{(x)}$ for $\\sigma \\in \\Sigma$, i.e.,\n\\[ \\Sigma_{(x)}=\\bigcup_{\\sigma\\in\\Sigma}\\sigma_{(x)}. \\]\nThen $\\Sigma_{(x)}$ is a subdivision of $\\Sigma$.\n\n\\medskip\n\nFor a pair of points $x_1, x_2$, we denote by $\\Sigma_{(x_1)(x_2)}$ the fan obtained by first blowing-up $\\Sigma$ at $x_1$, and then blowing-up the resulting fan at $x_2$. The definition extends to any ordered sequence of points $x_1, \\dots, x_k$ in $\\mathbb{R}^n$.\n\n\\begin{prop} \\label{prop:triangulation}\nLet $\\Sigma$ be a fan. Let $\\sigma_1, \\dots, \\sigma_k$ be the minimal faces of $\\Sigma$ which are not simplicial. For each such face $\\sigma_i$, $i\\in\\zint1k$, pick a vector $x_i$ in its interior. Then\n\\[ \\Sigma_{(x_1)(x_2)\\cdots(x_k)} \\]\nis a triangulation of $\\Sigma$.\n\\end{prop}\n\n\\begin{proof}\nFaces of $\\Sigma_{(x_1)}$ which were not in $\\Sigma$ are of the form $\\tau+\\mathbb{R}_+x_1$ where $\\tau\\in\\Sigma$ and $x_1\\not\\in\\TT\\tau$. Let $\\tau'=\\tau+\\mathbb{R}_+x_1$ be such a new face. Clearly $\\tau'$ is simplicial if and only if $\\tau$ is simplicial. By contrapositive, if $\\tau'$ is not simplicial, then $\\tau\\prec\\tau'$ is not simplicial. In any case, $\\tau'$ is not a non-simplicial minimal cone of $\\Sigma_{(x_1)}$. Therefore, the non-simplicial minimal cones of $\\Sigma_{(x_1)}$ are included in (in fact, equal to) $\\{\\sigma_2,\\dots,\\sigma_k\\}$. We concludes the proof by repeating this argument $k$-th time.\n\\end{proof}\n\n\n\\subsubsection{External cone over a polyhedral complex} We now define external cones over polyhedral complexes.\n\nFirst, let $\\phi\\colon\\mathbb{R}^n\\to\\mathbb{R}^m$ be an affine linear application and let $X$ be a polyhedral complex in $\\mathbb{R}^n$. The \\emph{image of $X$ by $\\phi$} denoted by $\\phi(X)$ is the polyhedral pseudo-complex $\\{\\phi(\\delta)\\st\\delta\\in X\\}$ of $\\mathbb{R}^m$. If the restriction $\\phi\\rest{\\supp X}$ is injective, $\\phi(X)$ is a polyhedral complex.\n\n\\medskip\n\nFor a subset $S$ of $\\mathbb{R}^n$, we denote by $\\adh{S}$ the topological closure of $S$ in $\\mathbb{R}^n$.\n\n\\begin{prop}[Coning over a polyhedron] \\label{prop:cone_polyedre}\nLet $\\delta$ be an arbitrary polyhedron which does not contain the origin in $\\mathbb{R}^n$. Let $H$ be any affine hyperplane containing $\\Tan\\delta$ but not $0$, and denote by $H_0$ the linear hyperplane parallel to $H$. Let $H_0^+$ be the half-space defined by $H_0$ which contains $\\delta$ in its interior. Then\n\\begin{enumerate}\n\\item\\label{enum:coning:1} $\\sigma:=\\adh{\\mathbb{R}_+\\delta}$ is a cone which verifies $\\delta=\\sigma\\cap H$ and $\\delta_\\infty=\\sigma\\cap H_0$.\n\\item\\label{enum:coning:2} Moreover, $\\sigma$ is the unique cone included in $H_0^+$ which verifies $\\delta=\\sigma\\cap H$.\n\\item\\label{enum:coning:3} In addition, if $\\delta$ is rational, resp. simplicial, then so is $\\sigma$.\n\\end{enumerate}\n\\end{prop}\nNote that by the terminology adapted in this paper, part \\ref{enum:coning:1} means $\\sigma$ is strongly convex, i.e., it does not contain any line.\n\\begin{proof} For part \\ref{enum:coning:1}, we will show more precisely that if\n\\[\\delta=\\conv(v_0,\\dots,v_l) +\\mathbb{R}_+u_1+\\dots+\\mathbb{R}_+u_m,\\]\nfor points $v_0, \\dots, v_l$ in $\\mathbb{R}^n$ and vectors $u_1, \\dots, u_m$ in $H_0$, then $\\sigma=\\sigma'$ where\n\\[\\sigma':=\\mathbb{R}_+v_0+\\dots+\\mathbb{R}_+v_l+\\mathbb{R}_+u_1+\\dots+\\mathbb{R}_+u_m.\\]\n\nNote that obviously we have $\\delta\\subset\\sigma'$, which implies the inclusion $\\sigma\\subseteq\\sigma'$. To prove the inclusion $\\sigma'\\subseteq\\sigma$, since $v_0,\\dots,v_l \\in \\sigma$, we need to show $u_1, \\dots, u_m \\in \\sigma$. Let $x$ be an element of $\\delta$. For all $\\lambda\\geq0$, all the points $x+\\lambda u_j$ are in $\\delta$. Dividing by $\\lambda$, we get $x\/\\lambda+u_j\\in\\sigma$. Making $\\lambda$ tend to infinity, we get $u_j\\in\\sigma$ for all $j=1, \\dots, m$. This proves $\\sigma = \\sigma'$.\n\nWe now show that $\\sigma$ is a cone, i.e., it does not contain any line. To show this, let $\\lambda_0,\\dots,\\lambda_l,\\mu_1,\\dots,\\mu_m\\in\\mathbb{R}_+$ such that $\\lambda_0v_0+\\dots+\\lambda_lv_l+\\mu_1u_1+\\dots+\\mu_mu_m=0$. It will be enough to show all the coefficients $\\lambda_j$ and $\\mu_i$ are zero.\n\nFirst, if one of the coefficients $\\lambda_j $ was non-zero, dividing by the linear combination above with the sum $\\sum_{j=1}^l \\lambda_j$, we would get $ 0 \\in \\delta$, which would be a contradiction. This shows all the coefficients $\\lambda_j$ are zero. Moreover, $\\delta_\\infty$ is a cone, i.e., it is strongly convex, which proves that all the coefficients $\\mu_i$ are zero as well. (Note that in our definition of polyhedra, we assume that a polyhedron is strongly convex, that is it does not contain any affine line.)\n\n\\medskip\n\nTo finish the proof of part \\ref{enum:coning:1}, it remains to show $\\delta = \\sigma \\cap H$ and that $\\delta_\\infty = \\sigma \\cap H_0$.\n\nObviously, we have the inclusion $\\delta\\subseteq\\sigma\\cap\\Tan\\delta\\subseteq\\sigma\\cap H$. Proceeding by absurd, suppose there exists $y\\in(\\sigma\\cap H)\\setminus\\delta$. We can write $y=\\lambda_0v_0+\\dots+\\lambda_lv_l+\\mu_1u_1+\\dots+\\mu_mu_m$ with non-negative coefficients $\\lambda_j, \\mu_i$.\n\nWe first show that all the coefficients $\\lambda_j$ are zero. Otherwise, let $\\Lambda=\\sum_i\\lambda_i$, which is non-zero. Since we are assuming $y\\not\\in\\delta$, we get $\\Lambda\\neq 1$. The point $x=y\/\\Lambda$ belongs to $\\delta$, and so both points $x,y\\in \\delta \\subset H$. It follows that any linear combination $(1-\\lambda) x +\\lambda y$ for $\\lambda\\in \\mathbb{R}$ belongs to $ H$. In particular,\n\\[  0 = \\frac\\Lambda{1-\\Lambda}x-\\frac1{1-\\Lambda}y\\in H, \\]\nwhich is a contradiction. This shows $\\lambda_0=\\dots =\\lambda_l =0$.\n\nWe thus infer that $y = \\mu_1 u_1 + \\dots+\\mu_m u_m$ belongs to $H_0$. But this is a contradiction because $H \\cap H_0 =\\emptyset$. We conclude with the equality $\\delta = \\sigma \\cap H$.\n\nWe are left to show $\\delta_\\infty=\\sigma\\cap H_0$. Take a point $y\\in\\sigma\\cap H_0$, and write $y=\\lambda_0v_0+\\dots+\\lambda_nv_n+\\mu_1u_1+\\dots+\\mu_mu_m$ with non-negative coefficients $\\lambda_j, \\mu_i$. Then by a similar reasoning as above, all the coefficients $\\lambda_i$ should be zero, which implies $y\\in\\delta_\\infty$. This implies the inclusion $\\sigma \\cap H_0 \\subseteq \\delta_\\infty$. The inclusion $\\delta_\\infty \\subset H_0$ comes from the inclusion $\\delta_\\infty \\subset \\TT\\delta \\subseteq H_0$ and the inclusion $\\delta_\\infty \\subset \\sigma$.\n\n\\medskip\n\nWe now prove part \\ref{enum:coning:2}. Let $\\sigma'$ be a cone included in $H_0^+$ such that $\\sigma'\\cap H=\\delta$. Obviously, $\\sigma=\\adh{\\mathbb{R}_+\\delta}\\subseteq\\sigma'$. To prove the inverse inclusion, let $y$ be a point $\\sigma'$. If $y\\not\\in H_0$, then there exists a scalar $\\lambda>0$ such that $\\lambda y\\in H$. It follows that $\\lambda y\\in\\delta$ and so $y\\in\\sigma$. If now $y\\in H_0$, then for all points $x\\in\\delta$ and any real $\\lambda>0$, the point $x+\\lambda y$ belongs to $\\sigma\\cap H$, which is equal to $\\delta$ by the first part of the proposition. We infer that $y\\in\\delta_\\infty\\subset\\sigma$. This implies that $\\sigma'=\\sigma$ and the unicity follows.\n\n\\medskip\n\nPart \\ref{enum:coning:3} follows from the description of the cone $\\sigma$ given in the beginning of the proof.\n\\end{proof}\n\nLet now $X$ be a polyhedral complex in $\\mathbb{R}^n$. We define the \\emph{cone over $X$} denoted by $\\cone{X}$ by\n\\[ \\cone{X}:=\\Bigl\\{\\,\\adh{\\mathbb{R}_+\\delta}\\st\\delta\\in X\\,\\Bigr\\}\\cup\\Bigl\\{\\conezero\\Bigr\\}. \\]\n\nIn particular, we note that this is not always a pseudo-fan. However, if one assumes that $\\supp{X}$ does not contain $0$, then $\\cone{X}$ is indeed a pseudo-fan which contains $X_\\infty$.\n\n\\medskip\n\nWe define the \\emph{external cone over $X$} denoted by $\\coneup{X}$ as the pseudo-fan of $\\mathbb{R}^{n+1}$ defined as follows. This is also called sometimes the \\emph{homogenization of $X$},\n\nLet $(e_0,\\dots,e_n)$ be the standard basis of $\\mathbb{R}^{n+1}$, and let $x_0,\\dots,x_n$ be the corresponding coordinates.\nConsider the projection $\\pi\\colon\\mathbb{R}^{n+1}\\to\\mathbb{R}^n$ defined by\n\\[ \\pi(x_0,\\dots,x_n)=(x_1,\\dots,x_n). \\]\n\nConsider the affine hyperplane $H_1$ of $\\mathbb{R}^{n+1}$ given by the equation $x_0=1$. The restriction of $\\pi$ to $H_1$ gives an isomorphism $\\pi_1\\colon H_1\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}}\\mathbb{R}^n$, and we define\n\\[ \\coneup X:=\\cone{\\pi_1^{-1}(X)}. \\]\n\nIn particular, for any polyhedron $\\delta$, we get the external cone\n\\[ \\coneup \\delta:=\\cone{\\pi_1^{-1}(\\delta)}. \\]\nBy Proposition~\\ref{prop:cone_polyedre}, intersecting with $H_1$ gives\n\\[ \\pi(\\coneup{X}\\cap H_1)=X. \\]\n\nThe pseudo-fan $\\coneup{X}$ is included in $\\{x_0\\geq 0\\}$. Moreover, denoting by $H_0$ the linear hyperplane $\\{x_0 = 0\\}$, we get\n\\[\\coneup{X}\\rest{H_0}=\\coneup{X}\\cap H_0.\\]\nApplying again Proposition~\\ref{prop:cone_polyedre}, we deduce that $\\pi(\\coneup{X}\\cap H_0)$ is the recession pseudo-fan $X_\\infty$ of $X$.\n\n\n\\subsubsection{From $X_\\infty$ to $X$ and vice-versa}\n\\label{sec:X_infty_X}\n\nLet $X$ be any polyhedral complex such that $X_\\infty$ is a fan. Let $f$ be a piecewise linear function on $X$. The function $f$ naturally induces a function on $\\cone(X)\\cap H_1$. This function can be extended by linearity to $\\cone{X}\\setminus H_0$. We denote this extension by $\\~f$. We say that \\emph{$f_\\infty$ is well-defined} if $\\~f$ can be extended by continuity to $H_0$. In this case, we define $f_\\infty$ to be the restriction of $\\~f$ to $X_\\infty=\\coneup{X}\\cap H_0$.\n\n\\begin{remark}\nNotice that if $f$ is strictly convex and if $f_\\infty$ is well-defined, then $f_\\infty$ is strictly convex.\n\\end{remark}\n\n\\medskip\n\nAssume $X$ is simplicial. In this case, recall that for any polyhedron $\\delta\\in X$ and any point $x$ in $\\delta$, there is a unique decomposition $x=x_\\f+x_\\infty$ with $x_\\f\\in\\delta_\\f$ and $x_\\infty\\in\\delta_\\infty$. We define the \\emph{projection onto the finite part} $\\pi_\\f\\colon\\supp{X}\\to\\supp{X_\\f}$ as the map $x\\mapsto x_\\f$. Similarly, we define the \\emph{projection onto the asymptotic part} $\\pi_\\infty\\colon \\supp X\\to \\supp{X_\\infty}$ by $x\\mapsto x_\\infty$. If $Y_\\f$ is subdivision of $X_\\f$, we define\n\\[ \\pi_\\f^{-1}(Y_\\f):=\\bigcup_{\\delta\\in X}\\Bigl\\{\\,\\gamma+\\sigma\\,\\st\\,\\sigma\\prec\\delta_\\infty, \\gamma\\in Y_\\f, \\gamma\\subset\\delta_\\f\\, \\Bigr\\}. \\]\n\n\\begin{remark} \\label{rem:extension_compatible}\nSetting $Z:=\\pi_\\f^{-1}(Y_\\f)$, one can show that $Z$ is a polyhedral complex, which is moreover simplicial if $Y_\\f$ is simplicial. In addition, we have $Z_\\f=Y_\\f$ et $Z_\\infty=X_\\infty$.\n\\end{remark}\n\n\n\n\\subsection{Regular subdivisions and convexity}\n\\label{subsec:convexite}\n\nLet now $X$ be a polyhedral complex in $\\mathbb{R}^n$, and let $\\delta$ be a face of $X$. Recall that a function $f$ is called strictly positive around $\\delta$ if there exists a a neighborhood $V$ of the relative interior of $\\delta$ such that $f$ vanishes on $\\delta$ and it is strictly positive on $V \\setminus \\delta$. Recall as well from Section~\\ref{sec:tropvar} that a function $f\\in\\lpm(X)$ is called \\emph{strictly convex around $\\delta$} if there exists a function $\\ell\\in\\aff(\\mathbb{R}^n)$ such that $f-\\ell$ is strictly positive around the relative interior of $\\delta$ in $\\supp{X}$. We also note that $\\K_+(X)$ denotes the set of strictly convex functions on $X$, i.e., the set of those functions in $\\lpm(X)$ which are strictly convex around any face $\\delta$ of $X$.\n\n\\begin{remark} For a polyhedral complex $X$, the set $\\K_+(X)$ is an open convex cone in the real vector space $\\lpm(X)$, i.e., it is an open set in the natural topology induced on $\\lpm(X)$, and moreover, if $f,g\\in\\K_+(X)$, then we have $f+g \\in\\K_+(X)$ and $\\lambda f \\in\\K_+(X)$ for any positive real number $\\lambda>0$. In addition, for $\\ell\\in\\aff(X)$, we have $f+\\ell\\in\\K_+(X)$.\n\\end{remark}\n\n\\begin{example} If $P$ is a polyhedron in $\\mathbb{R}^n$, then we have $0\\in\\K_+(\\face P)$. Indeed, for each face $\\tau$ of $P$, by definition of faces $\\delta$ in a polyhedron, there exists a linear form $\\ell$ on $\\mathbb{R}^n$ which is zero on $\\delta$ and which is strictly negative on $P\\setminus \\delta$, so $0-\\ell = -\\ell$ is strictly positive around $\\delta$.\n\\end{example}\n\nWe call a polyhedral complex $X$ \\emph{quasi-projective} if the set $\\K_+(X)$ is non-empty. Quasi-projective polyhedral complexes are sometimes called \\emph{convex} in the literature.\n\n\\begin{remark}\nIn the case of fans, the notion coincides with the notion of quasi-projectivity (convexity) used in the previous section.\n\\end{remark}\n\nWe are now going to make a list of operations on a polyhedral complex which preserves the quasi-projectivity property.\nTo start, we have the following which is straightforward.\n\n\\begin{prop} \\label{prop:sous-complexe_projectif}\nA subcomplex of a quasi-projective polyhedral complex is quasi-projective.\n\\end{prop}\n\nLet now $Y$ be a subdivision of $X$. For each face $\\delta$ of $X$, recall that the restriction $Y\\rest \\delta$ consists of all the faces of $Y$ which are included in $\\delta$.\n\n\\medskip\n\nLet $X$ be a polyhedral complex. A \\emph{regular subdivision of $X$} is a subdivision $Y$ such that there exists a function $f\\in\\lpm(Y)$ whose restriction to $Y\\rest\\delta$ for any face $\\delta\\in X$ is strictly convex. In this case, we say $Y$ is \\emph{regular relative to $X$}, and that the function $f$ is \\emph{strictly convex relative to $X$}. For example, a strictly convex function $f$ on $Y$ is necessary strictly convex relative to $X$ (but the inverse is not necessarily true).\n\n\\begin{remark} The terminology is borrowed from the theory of triangulations of polytopes and secondary polytopes~\\cites{GKZ, DRS}. In fact, a polyhedral subdivision of a polytope is regular in our sense if and only it is regular in the sense of~\\cite{GKZ}.\n\\end{remark}\n\n\\begin{remark} To justify the terminology, we should emphasize that regularity for a subdivision is a relative notion while quasi-projectivity for a polyhedral complex is absolute. In particular, the underlying polyhedral complex of a regular subdivision does not need to be quasi-projective.\n\\end{remark}\n\n\\begin{remark}\nA function $f \\in \\lpm(Y)$ which is strictly convex relative to $X$ allows to recover the subdivision $Y$ as the polyhedral complex whose collection of faces is\n\\[ \\bigcup_{\\delta\\in X}\\bigcup_{\\substack{\\ell\\in(\\mathbb{R}^n)^\\dual \\\\ \\ell\\rest\\delta\\leq f\\rest\\delta}}\\Bigl\\{x\\in\\delta\\st\\ell(x)=f(x)\\Bigr\\}\\ \\setminus\\Bigl\\{\\emptyset\\Bigr\\}. \\qedhere \\]\n\\end{remark}\n\n\\begin{prop} \\label{prop:projectivite_convexite}\nA regular subdivision of a quasi-projective polyhedral complex is itself quasi-projective.\n\\end{prop}\n\n\\begin{proof}\nLet $X$ be a quasi-projective polyhedral complex in $\\mathbb{R}^n$ and let $f\\in\\K_+(X)$. Let $Y$ be a regular subdivision of $X$ and let $g\\in\\lpm (Y)$ be strictly convex relative to $X$. We show that for any small enough $\\epsilon>0$, the function $f+ \\epsilon g \\in \\lpm(Y)$ is strictly convex on $Y$, which proves the proposition.\n\nLet $\\gamma$ be a face of $Y$. It will be enough to show that $f+ \\epsilon g$ is strictly convex around $\\gamma$ for any sufficiently small value of $\\epsilon>0$.\n\nLet $\\delta$ be the smallest face of $X$ which contains $\\gamma$. Let $\\ell\\in\\aff(X)$ and $\\ell' \\in \\aff(Y\\rest\\delta)$ be two affine linear functions such that $f-\\ell$ is strictly positive around $\\delta$ and $g\\rest{\\delta} -\\ell'$ is strictly positive around $\\gamma$. Without loss of generality, we can assume that $\\ell' \\in \\aff(X)$.\n\nLet $V$ be a small enough neighborhood of the interior of $\\gamma$ in $\\supp{X}$. Equipping $\\mathbb{R}^n$ with its Euclidean norm, we can in addition choose $V$ in such a way that for each point $x \\in V$, the closest point of $x$ in $\\delta$ belongs to $V$.\n\nSince $f$ and $g$ are piecewise linear, we can find $c, c'>0$ such that for any $x\\in V$,\n\\begin{gather*}\nf(x)-\\ell(x)\\geq c\\,\\dist(x,\\delta)\\quad\\text{and} \\\\\ng(x)-\\ell'(x)\\geq \\min_{y\\in V\\cap\\delta}\\big\\{g(y) -\\ell'(y)\\bigr\\}-c'\\,\\dist(x,\\delta)=-c'\\,\\dist(x,\\delta),\n\\end{gather*}\nwhere $\\dist(x,\\delta)$ denotes the Euclidean distance between the point $x$ and the face $\\delta$.\n\nWe infer that the function $f+\\epsilon g-(\\ell+\\epsilon \\ell')$ is strictly positive around $\\gamma$ for any value of $\\epsilon<c\/c'$. It follows that $f+\\epsilon g$ is strictly convex around $\\gamma$ for small enough values of $\\epsilon>0$, as required. This shows that $Y$ is quasi-projective.\n\\end{proof}\n\nAs an immediate consequence we get the following.\n\n\\begin{cor} \\label{cor:transitivite_projectivite}\nThe relation of \\enquote{being a regular subdivision} is transitive.\n\\end{cor}\n\n\\begin{proof}\nLet $Y$ be a regular subdivision of a polyhedral complex $X$ with $f \\in \\lpm(Y)$ strictly convex relative to $X$, and let $Z$ be a regular subdivision of $Y$ with $g \\in \\lpm(Z)$ strictly convex relative to $Y$.\nIt follows that for a face $\\delta \\in X$, the restriction $f\\rest{\\delta}$ is strictly convex on $Y\\rest{\\delta}$, and the restriction $g\\rest{\\delta}$, seen as a function in $\\lpm (Z\\rest\\delta)$, is strictly convex relative to $Y \\rest \\delta$.\n\nBy the previous proposition, more precisely by its proof, for sufficiently small values of $\\epsilon>0$, the function $f\\rest{\\delta}+\\epsilon g\\rest{\\delta}$ is strictly convex on $Z\\rest{\\delta}$. Taking $\\epsilon>0$ small enough so that this holds for all faces $\\delta$ of $X$, we get the result.\n\\end{proof}\n\n\\begin{prop} \\label{prop:eclate_projectif}\nLet $\\Sigma$ be a fan in $\\mathbb{R}^n$ and take a point $x\\in \\mathbb{R}^n\\setminus\\{0\\}$. The blow-up $\\Sigma_{(x)}$ of $\\Sigma$ at $x$ is regular relative to $\\Sigma$.\n\\end{prop}\n\n\\begin{proof} There exists a unique function $ f \\in \\lpm (\\Sigma_{(x)}) $ which vanishes on any face of $\\Sigma$ that does not contain $x$ and which takes value $ -1 $ at $ x $. We show that this function is strictly convex relative to $\\Sigma$, which proves the proposition.\n\nLet $\\sigma \\in \\Sigma$. If $\\sigma$ does not contain $x$, then $\\sigma$ belongs to $\\Sigma_{(x)}$, and $f$ is zero, therefore convex, on $\\Sigma_{(x)}\\rest\\sigma=\\face{\\sigma}$. Consider now the case where $ x \\in \\sigma $. Let $ \\tau $ be a face of $ \\sigma$ not containing $x$. By the definition of the blow-up, we have to show that $ f \\rest \\sigma$ is convex around $ \\tau $ and around $ \\tau + \\mathbb{R}_+ x $. Let $ \\ell \\in \\aff (\\mathbb{R}^n)$ be an affine linear function which vanishes on $\\tau $ and which is strictly negative on $ \\sigma \\setminus \\tau $. Moreover, we can assume without losing generality that $ \\ell(x) = -1 $. The function $f-\\ell$ is therefore zero on $\\tau + \\mathbb{R}_+ x $. In addition, if $\\nu$ is a face of $\\sigma$ not containing $x$, $f$ vanishes on $\\nu$, so $f-\\ell$ is strictly positive on $\\nu \\setminus \\tau$. If now $y$ is a point lying in $\\sigma \\setminus (\\tau + \\mathbb{R}_+ x)$, there is a face $\\nu$ of $\\sigma$ such that $ y$ belongs to $(\\nu \\setminus \\tau) + \\mathbb{R}_+ x$, so we can write $y = \\lambda x + z $ for some real number $\\lambda \\geq 0$ and an element $z \\in \\nu \\setminus \\tau$. Given the definition of $f$ and the choice of $\\ell$, we see that\n\\[ (f- \\ell) (y) = \\lambda \\cdot0 + (f- \\ell) (z)> 0. \\]\nThus, $f-\\ell$ is strictly positive around $\\tau + \\mathbb{R}_+ x$ in $\\sigma$. In the same way, we see that $f-2\\ell$ is strictly positive around $\\tau$ in $\\sigma$. So $f\\rest \\sigma$ is strictly convex around $\\tau + \\mathbb{R}_+ x$ and around $\\tau$ in $\\Sigma_{(x)}\\rest \\sigma$, and the proposition follows.\n\\end{proof}\n\n\\begin{prop} \\label{prop:polyedre_convexe}\nLet $P$ be a polytope in $\\mathbb{R}^n$ which contains $0$. The fan $\\Sigma:=\\cone{\\face{P}\\rest{\\mathbb{R}^n\\setminus\\{0\\}}}$ consisting of cones of the form $\\mathbb{R}_+\\sigma$ for $\\sigma$ a face of $P$ is quasi-projective.\n\\end{prop}\n\n\\begin{proof} We omit the proof that $\\Sigma$ is a fan. In order to prove the quasi-projectivity, let $\\tau$ be a face of $P$ which does not contain $0$. There exists an affine linear $\\ell^\\tau$ function on $\\mathbb{R}^n$ which takes value $0$ on $\\tau$ and which takes negative values on $P \\setminus \\tau$. Moreover, we can assume that $\\ell^\\tau(0)=-1$.\nThe restrictions $\\ell^\\tau\\rest{\\cone{\\tau}}$ glue together to define a piecewise affine linear function $f$ on $\\Sigma$. One can show that $f$ is strictly convex on $\\Sigma$.\n\\end{proof}\n\n\\begin{prop} \\label{prop:coupe_projective}\nLet $X$ be a polyhedral complex and let $H$ be an affine hyperplane in $\\mathbb{R}^n$. The hyperplane cut $X\\cdot H$ is regular relative to $X$.\n\\end{prop}\n\n\\begin{proof} Let $\\dist(\\,\\cdot\\,,\\cdot\\,)$ be the Euclidean distance in $\\mathbb{R}^n$. It is easy to see that the distance function $x\\mapsto \\dist(x, H)$ is strictly convex on $X\\cdot H$ relative to $X$.\n\\end{proof}\n\n\\begin{cor} \\label{prop:intersection_projective}\nSuppose that $X$ is quasi-projective. Then the intersection $X\\cap H$ is quasi-projective as well.\n\\end{cor}\n\n\\begin{proof} If $X$ is quasi-projective, by combining the preceding proposition with Proposition~\\ref{prop:projectivite_convexite}, we infer that the hyperplane cut $X\\cdot H$ is quasi-projective as well. Since $X\\cap H=(X\\cdot H)\\rest H$ is a polyhedral subcomplex of $X\\cdot H$, it will be itself quasi-projective by Proposition~\\ref{prop:sous-complexe_projectif}.\n\\end{proof}\n\n\\begin{prop} \\label{prop:extension_projectivite_partie_finie}\nLet $X$ be a polyhedral complex in $\\mathbb{R}^n$. If $X$ is simplicial and $Y_\\f$ is a regular subdivision of $X_\\f$, the subdivision $Y:=\\pi_\\f^{-1}(Y_\\f)$ of $X$ is regular relative to $X$. Moreover, there exists a function $f\\in\\lpm(Y)$ which is strictly convex relative to $X$ and such that $f_\\infty=0$.\n\\end{prop}\n\n\\begin{proof}\n\\renewcommand{\\int}{\\textrm{int}}\n\nBy Remark~\\ref{rem:extension_compatible}, the polyhedral complex $Y$ is a subdivision of $X$ whose finite part coincides with $Y_\\f$.\n\n\\medskip\n\nLet $f_\\f$ be a piecewise affine linear function on $Y_\\f$ which is strictly convex relative to $X_\\f$.\nLet $f:=\\pi_\\f^*(f_\\f) = f_\\f \\circ \\pi_\\f$ be the extension of $f$ to $\\supp Y=\\supp X$. It is clear that $f_\\infty=0$. For a face $\\gamma\\in Y$, since both the projection $\\pi_\\f\\rest\\gamma\\colon\\gamma\\to\\gamma_\\f$ and the restriction $f_\\f\\rest{\\gamma_\\f}$ are affine linear, $f\\rest\\gamma$ is also affine linear. This shows that $f \\in \\lpm(Y)$. We now show that $f$ is strictly convex relative to $X$, which proves the proposition.\n\n\\medskip\n\nLet $\\delta$ be a face of $X$ and let $\\gamma$ be a face in $Y$ which is included in $\\delta$. We need to show that $f\\rest\\delta$, as a piecewise linear function on $Y\\rest\\delta$, is strictly convex around $\\gamma$.\nBy the choice of $f_\\f$, we know that $f_\\f$ is strictly convex around $\\gamma_\\f$ in $Y_\\f\\rest{\\delta_\\f}$. Let $\\ell_\\f$ be an element of $\\aff(\\delta_\\f)$ such that $f_\\f\\rest{\\delta_\\f}-\\ell_\\f$ is strictly positive around $\\int(\\gamma_\\f)$ in $\\delta_\\f$, where $\\int(\\gamma_\\f)$ denotes the relative interior of $\\gamma_\\f$.\n\nThe function $\\ell:=\\pi_\\f^*(\\ell_\\f)$ is an element of $\\aff(\\delta)$. Moreover, the difference $f-\\ell$ is strictly positive around $\\int(\\gamma_\\f)+\\delta_\\infty$ in $\\delta$. Notice that the relative interior of $\\gamma$ is included in $\\int(\\gamma_\\f)+\\delta_\\infty$. Let $\\ell'\\in\\aff(\\delta)$ be an affine linear function which is zero on $\\gamma$ and which takes negative values on $\\bigl(\\gamma_\\f+\\delta_\\infty\\bigr)\\setminus\\gamma$. Then, for $\\epsilon>0$ small enough, $f-\\ell-\\epsilon\\ell'$ is strictly positive around $\\int(\\gamma)$. Thus, $f$ is strictly convex around $\\gamma$, and the proposition follows.\n\\end{proof}\n\n\n\n\\subsection{Unimodularity} In this section, we discuss the existence of unimodular subdivisions of a rational polyhedral complex in $\\mathbb{R}^n$.\n\n\\begin{prop} \\label{prop:subdivision_unimodulaire_projective}\nLet $\\Sigma$ be a rational simplicial fan in $\\mathbb{R}^n$. There exists a regular subdivision $\\~\\Sigma$ of $\\Sigma$ which is in addition unimodular with respect to $\\mathbb{Z}^n$. Moreover, we can assume that unimodular cones of $\\Sigma$ are still in $\\~\\Sigma$.\n\\end{prop}\n\n\\begin{proof}\nThis follows from the results of~\\cite{Wlo97}*{Section 9}, where it is shown how to get a unimodular subdivision $\\~\\Sigma$ of $\\Sigma$ by a sequence of blow-ups on some non-unimodular cones. By Proposition~\\ref{prop:eclate_projectif} and Corollary~\\ref{cor:transitivite_projectivite}, the resulting fan $\\~\\Sigma$ is a projective subdivision of $\\Sigma$, from which the result follows.\n\\end{proof}\n\nFor a rational vector subspace $W$ of $\\mathbb{R}^n$, we denote by $\\p_W\\colon\\mathbb{R}^n\\to\\rquot{\\mathbb{R}^n}{W}$ the resulting projection map. We endow the quotient $\\rquot{\\mathbb{R}^n}W$ by the quotient lattice $\\rquot{\\mathbb{Z}^n}{\\mathbb{Z}^n\\cap W}$.\n\n\\medskip\n\nLet now $P$ be a simplicial polyhedron in $\\mathbb{R}^n$ and $W:=\\TT(P_\\f)$. We say that \\emph{$P_\\infty$ is unimodular relative to $P_\\f$} if the projection $\\p_W(P_\\infty)$ is unimodular in $\\rquot{\\mathbb{R}^n}{W}$.\n\n\\begin{remark} \\label{rem:modulaire_subdivision}\nIf $P_\\infty$ is unimodular relative to $P_\\f$, then for any polytope $Q$ included in $\\TT(P_\\f)$, for the Minkowski sum $R:=Q+P_\\infty$ we see that $R_\\infty$ is unimodular relative $R_\\f$\n\\end{remark}\n\n\\begin{prop} \\label{prop:decomposition_unimodularite}\nLet $P$ be an integral simplicial polyhedron in $\\mathbb{R}^n$. Then $P$ is unimodular if and only if both the following conditions are verified:\n\\begin{enumerate}\n\\item $P_\\f$ is unimodular; and\n\\item $P_\\infty$ is unimodular relative to $P_\\f$.\n\\end{enumerate}\n\\end{prop}\n\nFor a collection of vectors $v_1, \\dots, v_m$ in a real vector space, we denote by $\\Vect(v_1, \\dots, v_m) := \\mathbb{R} v_1+ \\dots +\\mathbb{R} v_m$ the vector subspace generated by the vectors $v_i$.\n\\begin{proof}\nWe can write $P=\\conv(v_0,\\dots,v_l)+\\mathbb{R}_+v_{l+1}+\\dots+\\mathbb{R}_+v_m$, as in Section~\\ref{sec:recol}. Moreover, we can suppose without loss of generality that $v_0=0$, that $v_i$ is primitive in $\\mathbb{R}_+v_i$ for $i\\in\\zint{l+1,m}$, and that $\\TT(P)=\\mathbb{R}^n$. The claimed equivalence can be rewritten in the form: \\emph{$(v_1,\\dots,v_m)$ is a basis of $\\mathbb{Z}^n$ if and only if $(v_1,\\dots,v_l)$ is a basis of $\\mathbb{Z}^n\\cap\\Vect(v_1,\\dots,v_m)$ and $(v_{l+1},\\dots,v_m)$ is a basis of $\\rquot{\\mathbb{Z}^n}{\\mathbb{Z}^n\\cap\\Vect(v_1,\\dots,v_m)}$.} This is clear.\n\\end{proof}\n\n\\begin{cor} \\label{cor:coneup_modulaire}\nIf a polyhedron $P$ is \\emph{rational} and its coning $\\coneup{P}$ is unimodular in $\\mathbb{R}^{n+1}$, with respect to the lattice $\\mathbb{Z}^{n+1}$, then $P_\\infty$ is unimodular relative to $P_\\f$.\n\\end{cor}\n\n\\begin{proof}\nWe use the notations used in the definition of $\\coneup P$. So let $\\pi\\colon\\mathbb{R}^{n+1}\\to\\mathbb{R}^n$ be the projection, and let $H_0$ be the hyperplane $\\{x_0=0\\}$ and $\\pi_0:=\\pi\\rest{H_0}$. Let $u_1, \\dots, u_l, u_{l+1} \\dots, u_m\\in\\mathbb{R}^{n+1}$ be so that $\\coneup P=\\mathbb{R}_+u_1+\\dots+\\mathbb{R}_+u_m$ and $u_1,\\dots,u_l\\not\\in H_0$ while $u_{l+1}, \\dots, u_m\\in H_0$, and $u_1,\\dots,u_m$ is part of a basis of $\\mathbb{Z}^n$.\n\nWe apply the preceding proposition to $P':=\\conv(0,u_1,\\dots, u_l)+\\mathbb{R}_+u_{l+1}+\\dots+\\mathbb{R}_+u_m$ which is clearly unimodular. Setting $W'=\\TT(P'_\\f)$, we infer that $\\p_{W'}(P'_\\infty)$ is a unimodular cone in $\\rquot{\\mathbb{R}^{n+1}}{W'}$. Note that $\\rquot{\\mathbb{R}^{n+1}}{W'}=\\rquot{H_0}{W'\\!\\cap\\!H_0}$. Moreover, the linear application $\\pi_0\\colon H_0\\to\\mathbb{R}^n$ is an isomorphism which preserves the corresponding lattices, and which sends $P'_\\infty=\\coneup P\\cap H_0$ onto $P_\\infty$, and $W'\\cap H_0$ onto $W := \\TT(P_\\f)$. Therefore, $\\pi_0$ induces an isomorphism from $\\rquot{H_0}{W'\\!\\cap\\!H_0}$ to $\\rquot{\\mathbb{R}^n}{W}$ which sends $\\p_{W'}(P'_\\infty)$ onto $\\p_W(P_\\infty)$. This finally shows that $\\p_W(P_\\infty)$ is unimodular.\n\\end{proof}\n\n\\begin{thm} \\label{thm:triangulation_unimodulaire_projective}\nLet $X$ be a \\emph{compact} rational polyhedral complex in $\\mathbb{R}^n$ (so $X_\\infty=\\{\\conezero\\}$). There exists a positive integer $k$ and a regular subdivision $Y$ of $X$ such that $Y$ is unimodular with respect to the lattice $\\frac 1k \\mathbb{Z}^n$.\n\\end{thm}\n\n\\begin{proof}\nThis is a consequence of Theorem 4.1 of \\cite{KKMS}*{Chapter III}.\n\\end{proof}\n\n\\begin{remark} \\label{rem:stabilite_eventail_unimodulaire}\nNote that if a fan $\\Sigma$ in $\\mathbb{R}^n$ is unimodular with respect to $\\mathbb{Z}^n$, then it is as well unimodular with respect to any lattice of the form $\\frac1k\\mathbb{Z}^n$.\n\\end{remark}\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:triangulation_unimodulaire_convexe}}\nWe are now in position to prove the main theorem of this section. Let $\\Sigma_X:=\\coneup{X}$, which is a pseudo-fan, and define $H_0$, $H_1$ and $\\pi\\colon\\mathbb{R}^{n+1}\\to\\mathbb{R}^n$ as in the definition of the external cone. We thus have $X=\\pi(\\Sigma_X\\cap H_1)$ and $X_\\infty=\\pi(\\Sigma_X\\cap H_0)$.\n\n\\medskip\n\nWe first prove the existence of a regular subdivision of $\\Sigma_X$.\n\n\\medskip\n\nLet $P$ be a polytope containing $0$ in its interior. Define the fan $\\Sigma_1:=\\cone{\\face{P}\\rest{\\mathbb{R}^{n+1}\\setminus\\{0\\}}}$. This is a complete fan which moreover, by Proposition \\ref{prop:polyedre_convexe}, is projective. Define $\\Sigma_2$ as the fan obtained by slicing $\\Sigma_1$ with respect to a pencil of hyperplanes associated to $\\Sigma_X$. (We defined the slicing with respect to a fan but as mentioned previously, the definition extends to pseudo-fans.)\n\nBy Proposition \\ref{prop:decoupe_subdivision}, $\\Sigma_2$ contains a subdivision of $\\Sigma_X$.\nMoreover, since $\\Sigma_1$ is quasi-projective, and since by Proposition \\ref{prop:coupe_projective}, $\\Sigma_2$ is regular relative to $\\Sigma_1$ (being obtained by a sequence of hyperplane cuts), it follows that $\\Sigma_2$ is quasi-projective.\n\n\\medskip\n\nLet $\\Sigma_3:=\\Sigma_2\\cdot H_0$, and note that this is a regular subdivision of $\\Sigma_2$ by the same proposition. In particular, $\\Sigma_3$ itself is quasi-projective. The fan $\\Sigma_3$ is special in that if $\\Sigma'$ if a subdivision of $\\Sigma_3$, then any cone in $\\Sigma'$ is included in one of the two half-spaces $H_0^+$ or $H_0^-$ defined by $H_0$.\n\n\\medskip\n\nLet $\\Sigma_4$ be a rational triangulation of $\\Sigma_3$ obtained from $\\Sigma_3$ by applying Proposition \\ref{prop:triangulation}. Note that $\\Sigma_4$ is obtained from $\\Sigma_3$ by blow-ups, thus, by Proposition \\ref{prop:eclate_projectif}, it is regular relative to $\\Sigma_3$. In particular, it is quasi-projective.\n\n\\medskip\n\nLet $\\Sigma_5$ be a unimodular subdivision of $\\Sigma_4$, which exists by Proposition~\\ref{prop:subdivision_unimodulaire_projective}. The fan $\\Sigma_5$ is therefore unimodular and quasi-projective, and it contains moreover a subdivision of $\\Sigma_X$.\n\n\\medskip\n\nWe now use $\\Sigma_5$ to get a subdivision of $X$. Let $X_1:=\\pi(\\Sigma_5\\cap H_1)$.\n\n\\medskip\n\nSince $\\Sigma_5$ is quasi-projective, the intersection $\\Sigma_5\\cap H_1$ is quasi-projective as well. Since $\\Sigma_5$ contains a subdivision of $\\Sigma_X$, it follows that $\\Sigma_5\\cap H_1$ contains a subdivision of $\\Sigma_X\\cap H_1=X$.\n\nPassing now to the projection, we infer that $X_1$ is quasi-projective and contains a subdivision of $X$. More precisely, there exists a strictly convex function $f$ on $X_1$ such that $f_\\infty$ is well-defined.\n\n\\medskip\n\nPick $\\delta$ in $X_1$, and let $\\sigma$ be a cone in $\\Sigma_5$ such that $\\pi(\\sigma\\cap H_1)=\\delta$. Applying Proposition~\\ref{prop:cone_polyedre} to $\\delta':=\\sigma\\cap H_1$ and to the hyperplane $H_1$, given that $\\sigma$ is included in the half-space defined by $H_0$ which contains $\\delta'$, we infer that $\\sigma=\\adh{\\mathbb{R}_+\\delta}=\\coneup\\delta$. Since $\\sigma$ is unimodular, by Corollary~\\ref{cor:coneup_modulaire} we infer that $\\delta_\\infty$ is unimodular relative $\\delta_\\f$.\n\n\\medskip\n\nLet $X_2:=X_1\\rest{\\supp X}$. It follows that $X_2$ is a subdivision of $X$ which is rational, simplicial, and quasi-projective, and moreover, for any $\\delta\\in X_2$, the cone $\\delta_\\infty$ is unimodular relative to $\\delta_\\f$.\n\n\\medskip\n\nAt this moment, we are left to show the existence of a unimodular regular subdivision $Y$ of $X_2$. By Theorem~\\ref{thm:triangulation_unimodulaire_projective}, there exists an integer $k\\geq1$ and a regular subdivision $Y_\\f$ of $X_{2,\\f}$ such that $Y_\\f$ is unimodular with respect to the lattice $\\frac1k\\mathbb{Z}^n$. Define $Y:=\\pi_\\f^{-1}(Y_\\f)$. By Proposition~\\ref{prop:extension_projectivite_partie_finie}, $Y$ is regular relative to $X_2$, and so by quasi-projectivity of $X_2$, it is quasi-projective itself. More precisely, there exists a function $f$ strictly convex on $Y$ such that $Y_\\infty$ is well-defined.\n\n\\medskip\n\nUsing Remark~\\ref{rem:modulaire_subdivision}, we infer that for each face $\\delta \\in Y$, the cone $\\delta_\\infty$ is unimodular relative to $\\delta_\\f$, with respect to the lattice $\\mathbb{Z}^n$ of $\\mathbb{R}^n$. By Remark~\\ref{rem:stabilite_eventail_unimodulaire}, this remains true for the lattice $\\frac 1k\\mathbb{Z}^n$ as well.\n\n\\medskip\n\nApplying Proposition~\\ref{prop:decomposition_unimodularite}, we finally conclude that $Y$ is unimodular with respect to the lattice $\\frac 1k\\mathbb{Z}^n$. At present, we have obtained a regular subdivision $Y$ of $X$ which is moreover quasi-projective and unimodular with respect to the lattice $\\frac 1k \\mathbb{Z}^n$. Moreover, it is clear that $Y_\\infty$ is a fan. This finishes the proof of Theorem~\\ref{thm:triangulation_unimodulaire_convexe}.\n\n\\medskip\n\n\n\n\\subsection{Unimodular triangulations preserving the recession fan}\nWe finish this section by presenting the proof of Theorem~\\ref{thm:unimodular_preserving_recession}. We use the same notations as in the previous section for external cones.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:unimodular_preserving_recession}]\nLet $\\Sigma_1=\\coneup X$. We know that $\\Sigma_1\\rest{H_0}=X_\\infty$ is a unimodular fan. Thus $\\Sigma_1$ is a fan.\n\n\\medskip\n\nBy Proposition~\\ref{prop:triangulation}, we can construct a simplicial subdivision $\\Sigma_2$ of $\\Sigma_1$. Moreover, since faces of $\\Sigma_1\\rest{H_0}$ are already simplicial, the construction described in Proposition~\\ref{prop:triangulation} does not change this part of the fan, i.e., $\\Sigma_2\\rest{H_0}=\\Sigma_1\\rest {H_0}$.\n\n\\medskip\n\nIn the same way, by Proposition~\\ref{prop:subdivision_unimodulaire_projective}, there exists a unimodular subdivision $\\Sigma_3$ of $\\Sigma_2$. Moreover, since $\\Sigma_2\\rest{H_0}$ is unimodular, the same proposition ensures we can assume $\\Sigma_3\\rest{H_0}=\\Sigma_2\\rest{H_0}$.\n\n\\medskip\n\nSet $X_1=\\pi(\\Sigma_3\\cap H_1)$. Then $X_1$ is a triangulation of $X$ with $X_{1,\\infty}=X_\\infty$.\n\\medskip\n\nBy Theorem~\\ref{thm:triangulation_unimodulaire_projective}, there exists an integer $k\\geq1$ and a subdivision $Y_\\f$ of $X_{1,\\f}$ such that $Y_\\f$ is unimodular with respect to the lattice $\\frac1k\\mathbb{Z}^n$. Define $Y:=\\pi_\\f^{-1}(Y_\\f)$. Clearly, $Y_\\infty=X_{1,\\infty}=X_\\infty$.\n\n\\medskip\n\nUsing the same argument as in the final part of the proof of Theorem~\\ref{thm:triangulation_unimodulaire_convexe} in the previous section, we infer that $Y$ is unimodular, which concludes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Tropical Steenbrink spectral sequence}\\label{sec:steenbrink}\n\nLet $V:=\\mathbb{R}^n$ be a vector space. Denote by $N$ the lattice $\\mathbb{Z}^n\\subset V$, thus $V=N_\\mathbb{R}$. Let $Y$ be a unimodular rational simplicial complex of dimension $d$ in $N_\\mathbb{R}$ whose support $\\supp{Y}$ is a smooth tropical variety. Assume that the recession pseudo-fan of $Y$ is a fan $Y_\\infty$. This fan induces a partial compactification $\\TP_{Y_\\sminfty}$ of $\\mathbb{R}^n$. Denote by $X$ the extended polyhedral complex induced by $Y$ on the closure of $Y$ in $\\TP_{Y_\\sminfty}$. Then, $\\supp{X}$ is a smooth compact tropical variety. To simplify the notations, we will denote by $X_\\infty$ the recession fan $Y_\\infty$ of $Y$, and by $\\TP_{X_{\\sminfty}}$ the partial compactification $\\TP_{Y_{\\sminfty}}$ of $\\mathbb{R}^n$.\n\n\\medskip\n\nRecall from Section~\\ref{sec:tropvar} that the space $\\TP_{X_\\sminfty}$ is naturally stratified by open strata $N^\\sigma_\\mathbb{R}$ for $\\sigma\\in X_\\infty$. If $\\sigma\\in X_\\infty$, we denote by $X^\\sigma$ the intersection of $X$ with $N^\\sigma_\\mathbb{R}$. Notice that $X^\\sigma$ is a unimodular simplicial complex of dimension $d-\\dims{\\sigma}$. In particular, $X^\\conezero=Y$. If $\\delta\\in X$ is a face, we denote by $\\sed(\\delta)\\in X_\\infty$ its sedentarity: this is the unique cone $\\sigma\\in X_\\infty$ such that the relative interior of $\\delta$ is in $N^\\sigma_\\mathbb{R}$.\n\n\\medskip\n\nRecall that for a simplicial polyhedron $\\delta$ in $\\mathbb{R}^n$, we defined $\\delta_\\infty$ and $\\delta_\\f$ in Section~\\ref{sec:tropvar}. We extend the definition to faces $\\delta$ of $X$ of higher sedentarity as follows. By $\\delta_\\infty$ (resp.\\  $\\delta_\\f$) we mean $(\\delta\\cap N^{\\sed(\\delta)}_\\mathbb{R})_\\infty$ (resp.\\  $(\\delta\\cap N^{\\sed(\\delta)}_\\mathbb{R})_\\f$) which belongs to the fun $(X^{\\sed(\\delta)})_\\infty$ (resp.\\  $(X^{\\sed(\\delta)})_\\f$).\n\n\\medskip\n\nThe \\emph{max-sedentarity} of a face $\\delta \\in X$ that we denote by $\\maxsed(\\delta)$ is defined as\n\\[\\maxsed(\\delta) :=\\max\\Bigl\\{\\sed(\\gamma) \\st \\gamma \\prec \\delta\\Bigr\\}.\\]\nOne verifies directly that we have $\\maxsed(\\delta) = \\sed(\\delta)+\\delta_\\infty$ and that this is the maximum cone $\\sigma\\in X_\\infty$ such that $\\delta$ intersects $N^\\sigma_\\mathbb{R}$.\n\n\\medskip\n\nFor each $\\delta \\in X$, let $\\Sigma^\\delta$ be the fan in $\\rquot{N^{\\sed(\\delta)}_\\mathbb{R}\\!}{\\TT\\delta}$ induced by the star of $\\delta$ in $X$. Unimodularity of $X$ implies that the fan $\\Sigma^\\delta$ is unimodular. We denote as before by $\\comp\\Sigma^\\delta$ the canonical compactification of $\\Sigma^\\delta$.\n\n\\medskip\n\nWe define the \\emph{$k$-th cohomology} $H^{k}(\\delta)$ of $\\comp\\Sigma^\\delta$ by\n\\[H^k(\\delta):= \\bigoplus_{r+s = k} H_{\\trop}^{r,s}(\\comp \\Sigma^\\delta) = \\begin{cases} H_{\\trop}^{k\/2,k\/2} (\\comp \\Sigma^\\delta) \\simeq A^{k\/2}(\\Sigma^\\delta) & \\textrm{ if $k$ is even} \\\\\n0 & \\textrm{ otherwise}\n\\end{cases}\\]\nwhere the last equalities follows from Theorem~\\ref{thm:HI}, proved in~\\cite{AP}.\n\n\n\n\\subsection{Basic maps} \\label{sec:basic_maps}\n\nWe define some basic maps between the cohomology groups associated to faces of $X$.\n\n\n\\subsubsection{The sign function} We will need to make the choice of a \\emph{sign function} for $X$. This is a function\n\\[\\sign\\colon \\Bigl\\{(\\gamma,\\delta)\\in X\\times X   \\,\\st\\, \\gamma\\ssubface\\delta \\Bigr\\}\\longrightarrow \\{-1,+1\\}\\]\nwhich takes a value $\\sign(\\gamma, \\delta)\\in\\{-1,+1\\}$ for a pair of faces $\\gamma,\\delta\\in X$ with $\\gamma\\ssubface\\delta$, and which verifies the following property that we call the \\emph{diamond property}.\n\n\\begin{defi}[Diamond property for a sign function]\\rm Let $\\delta$ be a face of dimension at least 2 in $X$ and $\\nu$ be a face of codimension 2 in $\\delta$. The diamond property of polyhedra ensures that there exist exactly two different faces $\\gamma, \\gamma'\\in X$ such that $\\nu\\ssubface\\gamma,\\!\\gamma'\\ssubface\\delta$. Then, we require the sign function to verify the following compatibility\n\\[ \\sign(\\nu, \\gamma)\\sign(\\gamma, \\delta)=-\\sign(\\nu, \\gamma')\\sign(\\gamma', \\delta). \\]\nMoreover, if $e$ is an edge in $X$ and if $u$ and $v$ are its extremities, then we require the sign function to verify\n\\[ \\sign(u,e)=-\\sign(v,e). \\]\n\\end{defi}\nFor the existence, note that such a sign function can be easily defined by choosing an orientation on each face.\n\nFrom now on, we assume that a sign function for $X$ has been fixed once for all.\n\n\n\\subsubsection{The restriction and Gysin maps}\nIf $\\gamma$ and $\\delta$ are two faces with $\\gamma\\ssubface\\delta$, we have the \\emph{Gysin map}\n\\[ \\gys_{\\delta\\ssupface\\gamma}\\colon H^k(\\delta)\\to H^{k+2}(\\gamma), \\]\nand the \\emph{restriction map}\n\\[ \\i^*_{\\gamma\\ssubface\\delta}\\colon H^k(\\gamma)\\to H^k(\\delta), \\]\nwhere we define both maps to be zero if $\\gamma$ and $\\delta$ do not have the same sedentarity.\nIf $\\gamma$ and $\\delta$ have the same sedentarity, these maps correspond to the corresponding maps in the level of Chow groups defined in Section~\\ref{sec:local}.\n\n\\begin{remark} We recall that if $\\gamma$ and $\\delta$ have the same sedentarity, using the isomorphism of the Chow groups with the tropical cohomology groups, the two maps have the following equivalent formulation. First, we get an inclusion $\\comp\\Sigma^\\delta \\hookrightarrow \\comp \\Sigma^\\gamma$. To see this, note that $\\delta$ gives a ray $\\rho_{\\delta\/\\gamma}$ in $\\Sigma^\\gamma$. The above embedding identifies $\\comp\\Sigma^\\delta$ with the closure of the part of sedentarity $\\rho_{\\delta\/\\gamma}$ in $\\comp \\Sigma^\\gamma$.\nThe restriction map $\\i^*_{\\gamma\\ssubface\\delta}$ is then the restriction map $H^k(\\comp \\Sigma^\\gamma) \\to H^k(\\comp\\Sigma^\\delta)$ induced by this inclusion. The Gysin map is the dual of the restriction map $\\i^*_{\\gamma\\ssubface\\delta} \\colon H^{2d-2\\dims{\\delta}-k}(\\comp \\Sigma^\\gamma) \\to H^{2d - 2\\dims{\\delta}-k}(\\comp\\Sigma^\\delta)$, which by Poincar\\'e duality for $\\comp \\Sigma^\\delta$ and $\\comp \\Sigma^\\gamma$, is a map from $H^k(\\comp \\Sigma^\\delta)$ to $H^{k+2}(\\comp \\Sigma^\\gamma)$.\n\\end{remark}\n\nIn addition to the above two maps, if two faces $\\gamma$ and $\\delta$ do not have the same sedentarity, the projection map $\\pi_{\\delta\\ssupface\\gamma} \\colon N^{\\sed(\\delta)}_\\mathbb{R} \\to N^{\\sed(\\gamma)}_\\mathbb{R}$\ninduces an isomorphism $\\Sigma^\\delta \\simeq \\Sigma^\\gamma$, which leads to an isomorphism of the canonical compactifications, and therefor of the corresponding cohomology groups that we denote by\n\\[ \\pi^*_{\\gamma\\ssubface\\delta}\\colon H^k(\\gamma)\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} H^k(\\delta). \\]\n\n\\begin{remark} All the three above maps can be defined actually even if $\\gamma$ is not of codimension 1 in $\\delta$. Note that in this case, the Gysin map is a map $\\gys_{\\delta \\succ \\gamma} \\colon H^k(\\delta)\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} H^{k+ 2\\dims{\\delta} - 2\\dims{\\gamma}}(\\gamma)$.\n\\end{remark}\n\nLet now $S$ and $T$ be two collections of faces of $X$. We can naturally define the map\n\\[ \\begin{array}{rccc}\ni_{S,T}^*\\colon & \\bigoplus_{\\gamma\\in S}H^k(\\gamma) & \\longrightarrow & \\bigoplus_{\\delta\\in T}H^k(\\delta), \\\\\n\\end{array} \\]\nby sending, for any $\\gamma\\in S$, any element $x$ of $H^k(\\gamma)$ to the element\n\\[ \\i_{S,T}^* (x):=\\sum_{\\delta\\in T \\\\ \\delta\\ssupface\\gamma}\\sign(\\gamma, \\delta)\\i^*_{\\gamma\\ssubface\\delta}(x), \\]\nand extend it by linearity to the direct sum $\\bigoplus_{\\gamma\\in S} H^k(\\gamma)$.\n\nIn the same way, we define\n\\[ \\begin{array}{rccc}\n\\gys_{S,T}\\colon & \\bigoplus_{\\delta\\in S}H^k(\\delta) &  \\longrightarrow & \\bigoplus_{\\gamma \\in T}H^{k+2}(\\gamma), \\\\\n\\end{array} \\]\nby setting for $x \\in H^k(\\delta)$, for $\\delta \\in S$,\n\\[ \\gys_{S,T}(x) := \\sum_{\\gamma \\in T \\\\ \\delta \\ssupface \\gamma} \\sign(\\gamma,\\delta)\\gys_{\\delta \\ssupface \\gamma}(x). \\]\nSimilarly, we get\n\\[ \\begin{array}{rccc}\n\\pi^*_{S,T}\\colon & \\bigoplus_{\\gamma\\in S}H^k(\\gamma) &  \\longrightarrow & \\bigoplus_{\\delta\\in T}H^k(\\delta). \\\\\n\\end{array} \\]\n\nIn what follows, we will often extend maps in this way. Moreover, we will drop the indices if they are clear from the context.\n\n\n\n\\subsection{Combinatorial Steenbrink and the main theorem}\n\nDenote by $X_\\f$ the set of faces of $X$ whose closures do not intersect the boundary of $\\TP_{X_\\sminfty}$, i.e., the set of compact faces of $X_\\conezero$. Note that this was previously denoted by $X_{\\conezero,\\f}$ that we now simplify to $X_\\f$ for the ease of presentation.\n\nIn what follows, inspired by the shape of the first page of the Steenbrink spectral sequence~\\cite{Ste76}, we define bigraded groups $\\ST_1^{a,b}$ with a collection of maps between them. In absence of a geometric framework reminiscent to the framework of degenerating families of smooth complex projective varieties, the heart of this section is devoted to the study of this combinatorial shadow in order to obtain interesting information about the geometry of $X$.\n\n\\medskip\n\nFor all pair of integers $a, b \\in \\mathbb Z$, we define\n\\[ \\ST_1^{a,b} := \\bigoplus_{s \\geq \\dims{a} \\\\ s \\equiv a \\pmod 2} \\ST_1^{a,b,s} \\]\nwhere\n\\[ \\ST_1^{a,b,s} = \\bigoplus_{\\delta \\in X_\\f \\\\ \\dims\\delta =s} H^{a+b-s}(\\delta). \\]\n\nThe bigraded groups $\\ST_1^{a,b}$ come with a collection of maps\n\\[\\i^{a,b\\,*} \\colon \\ST^{a,b}_1 \\to \\ST_1^{a+1, b} \\qquad \\textrm{and} \\qquad \\gys^{a,b}\\colon \\ST^{a,b}_1 \\to \\ST_1^{a+1, b},\\]\nwhere both these maps are defined by our sign convention introduced in the previous section.\n\nIn practice, we drop the indices and denote simply by $\\i^*$ and $\\gys$ the corresponding maps.\n\\begin{remark} More precisely, we could view the collection of maps $\\i^*$ and $\\gys$ bi-indexed by $a,b$ as maps of bidegree $(1,0)$\n\\[\\i^* = \\bigoplus_{a,b}\\i^{a,b\\,*} \\colon \\bigoplus_{a,b} \\ST_1^{a,b} \\longrightarrow \\bigoplus_{a,b} \\ST_1^{a,b},\\]\nand\n\\[\\gys = \\bigoplus_{a,b}\\gys^{a,b} \\colon \\bigoplus_{a,b} \\ST_1^{a,b} \\longrightarrow \\bigoplus_{a,b} \\ST_1^{a,b}. \\qedhere \\]\n\\end{remark}\n\n\\begin{prop}\\label{prop:app1}\nThe two collections of maps $\\i^*$ and $\\gys$ have the following properties.\n\\begin{align*}\n\\i^* \\circ \\i^* =0, \\qquad \\gys \\circ \\gys =0, \\qquad \\i^* \\circ \\gys + \\gys \\circ \\i^* =0.\n\\end{align*}\n\\end{prop}\n\\begin{proof}\nSection~\\ref{sec:proofapp1} is devoted to the proof of this proposition.\n\\end{proof}\n\n\\medskip\n\nWe now define the \\emph{differential} $\\d\\colon \\ST_1^{a,b} \\to \\ST_1^{a+1,b}$ as the sum $\\d = \\i^*+ \\gys$. It follows from the properties given in the above proposition that we have the following.\n\n\\begin{prop}\nFor a unimodular triangulation of $\\mathfrak X$ and for any integer $b$, the differential $\\d$ makes $\\ST_1^{\\bul,b}$ into a cochain complex.\n\\end{prop}\n\n\\begin{remark} This proposition suggests $\\ST_1^{\\bul, \\bul}$ might be the first page of a spectral sequence converging to the cohomology of $X$. We could think that such a spectral sequence could be defined if we could \\emph{deform infinitesimally the tropical variety}. In the sequel we will only use the property stated in the proposition, that the lines of $\\ST_1^{\\bul, \\bul}$ with the differential $d$ form a cochain complex, as well as certain combinatorial properties of these cochain complexes.\n\\end{remark}\n\nFor a cochain complex $(C^\\bul, \\d)$, denote by $H^a(C^\\bul, \\d)$ its $a$-th cohomology group, i.e.,\n\\[H^a(C^\\bul, \\d) = \\frac{\\ker\\Bigl(\\d\\colon \\, C^{a} \\rightarrow C^{a+1}\\Bigr)}{\\Im \\Bigl(\\d\\colon \\, C^{a-1} \\rightarrow C^{a}\\Bigr)}. \\]\n\nIn this section we prove the following theorem, generalizing the main theorem of~\\cite{IKMZ} from approximable setting (when $X$ arises as the tropicalization of a family of complex projective varieties) to any general smooth tropical variety.\n\\begin{thm}[Steenbrink-Tropical comparison theorem] \\label{thm:steenbrink}\nThe cohomology of $(\\ST_1^{\\bul,b},\\d)$ is described as follows. For $b$ odd, all the terms $\\ST_1^{a,b}$ are zero, and the cohomology is vanishing. For $b$ even, let $b=2p$ for $p \\in \\mathbb Z$, then for $q\\in \\mathbb Z$, we have\n\\[H^{q-p}(\\ST^{\\bul,2p}_1, \\d) = H^{p,q}_{\\trop}(\\mathfrak X).\\]\n\\end{thm}\n\nThe rest of this section is devoted to the proof of Theorem~\\ref{thm:steenbrink}.\n\nIn absence of a geometric framework reminiscent to the framework of degenerating families of smooth complex projective varieties, the heart of our proof is devoted to introducing and developing certain tools, constructions, and results about the combinatorial structure of the Steenbrink spectral sequence.\n\n\\medskip\n\nBroadly speaking, the proof which follows is inspired by the one given in the approximable case~\\cite{IKMZ}, as well as by the ingredients in Deligne's construction of the mixed Hodge structure on the cohomology of a smooth algebraic variety~\\cite{Deligne-Hodge2} and by Steenbrink's construction of the limit mixed Hodge structure on the special fiber of a semistable degeneration~\\cite{Ste76}, both based on the use of the sheaf of logarithmic differentials and their corresponding spectral sequences.\n\nIn particular, we are going to prove first an analogous result in the tropical setting of the Deligne exact sequence which gives a resolution of the coefficient groups $\\SF^p$ with cohomology groups of the canonically compactified fans $\\comp \\Sigma^\\delta$. The proof here is based on the use of Poincar\\'e duality and our Theorem~\\ref{thm:HI}, the tropical analog of Feichtner-Yuzvinsky theorem~\\cite{FY}, which provides a description of the tropical cohomology groups of the canonically compactified fans. We then define a natural filtration that we call the \\emph{tropical weight filtration} on the coefficient groups $\\SF^p(\\cdot)$, inspired by the weight filtration on the sheaf of logarithmic differentials, and study the corresponding spectral sequence. The resolution of the coefficient groups given by the Deligne spectral resolution gives a double complex which allows to calculate the cohomology of the graded cochain complex associated to the weight filtration. We then show that the spectral sequence associated to the double complex corresponding to the weight filtration which abuts to the tropical cohomology groups, abuts as well to the cohomology groups of the Steenbrink cochain complex, which concludes the proof. The proof of this last result is based on the use of our Spectral Resolution Lemma~\\ref{lem:spectral_resolution} which allows to make a bridge between different spectral sequences.\n\n\n\n\\subsection{Tropical Deligne resolution}\n\nLet $\\Sigma$ be a unimodular fan structure on the support of a Bergman fan. We follow the notations of the previous section. In particular, for each cone $\\sigma$, we denote by $\\Sigma^\\sigma$ the induced fan on the star of $\\sigma$ in $\\Sigma$, and by $\\comp \\Sigma^\\sigma$ its canonical compactification.\n\n\\begin{thm}[Tropical Deligne resolution]\\label{thm:deligne}\nWe have the following exact sequence of $\\mathbb Q$-vector spaces:\n\\[0 \\rightarrow \\SF^p(\\conezero) \\rightarrow \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =p} H^0(\\sigma) \\rightarrow \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =p-1} H^2(\\sigma) \\rightarrow \\dots \\rightarrow \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =1} H^{2p-2}(\\sigma) \\rightarrow H^{2p} (\\conezero) \\to 0, \\]\nwhere the maps between cohomology groups are given by $\\gys$.\n\\end{thm}\n\n\\begin{remark} We should justify the name given to the theorem. In the realizable case when $\\Sigma$ has the same support as the Bergman fan $\\Sigma_\\Ma$ of a matroid $\\Ma$ realizable over the field of complex numbers, this can be obtained from the Deligne spectral sequence which describes the mixed Hodge structure on the cohomology of a complement of complex hyperplane arrangement. This is done in~\\cite{IKMZ}. Our theorem above states this is more general and holds for any Bergman fan.\n\\end{remark}\n\\begin{remark} The theorem suggests this should be regarded as a cohomological version of the inclusion-exclusion principle. The cohomology groups are described in terms of the coefficient sheaf and the coefficient sheaf can be recovered from the cohomology groups. It would be certainly interesting to see whether a cohomological version of the M\\\"obius inversion formula could exist.\n\\end{remark}\n\nThe rest of this section is devoted to the proof of this theorem. First using the Poincar\\'e duality for canonical compactifications $\\comp \\Sigma^\\sigma$, it will be enough to prove the exactness of the following complex for each $k$ (here $k=d-p$):\n\n\\begin{equation}\n0 \\rightarrow H^{2k}(\\conezero) \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =1} H^{2k}(\\sigma) \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =2} H^{2k}(\\sigma) \\to \\dots \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =d-k} H^{2k}(\\sigma) \\to \\SF_{d-k}(\\conezero) \\to 0.\n\\end{equation}\n\nRecall from Section~\\ref{sec:tropvar} that for any tropical variety $Z$, we denote by $\\Omega^k_Z$ the sheaf of tropical holomorphic $k$-forms on $Z$, defined as the kernel of the second differential operator from Dolbeault $(k,0)$-forms to Dolbeault $(k,1)$-forms on $Z$. Since we are going to use some results from~\\cite{JSS}, we note that this sheaf is denoted $\\mathcal L^k_Z$ in \\emph{loc. cit.}\n\n\\medskip\n\nWe will need the following alternative characterization of this sheaf, which also shows that it can be defined over $\\mathbb{R}$ as the sheafification of the combinatorial sheaf $\\SF^k$. Let $Z$ be a tropical variety with an extended polyhedral structure as in Section~\\ref{sec:tropvar}. If $U$ is an open set of $Z$, we say that $U$ is \\emph{nice} if either $U$ is empty, or there exists a face $\\gamma$ of $Z$ intersecting $U$ such that, for each face $\\delta$ of $Z$, every connected components of $U\\cap\\delta$ contains $U\\cap\\gamma$. (Compare with the \\emph{basic open sets} from~\\cite{JSS}.) This condition implies that $U$ is connected, and for each face $\\delta\\in Z$ which intersects $U$, $\\gamma$ is a face of $\\delta$ and $\\delta\\cap U$ is connected. We call $\\gamma$ the \\emph{minimum face of $U$}. Nice open sets form a basis of open sets on $Z$.\n\nThe sheaf $\\Omega^k_Z$ is then the unique sheaf on $Z$ such that, for each nice open set $U$ of $Z$ with minimum face $\\gamma$, we have\n\\[ \\Omega^k_Z(U)=\\SF^k(\\gamma). \\]\n\n\\medskip\n\nSuppose now $Z \\subseteq \\comp Z$ is a compactification of $Z$. We denote by $\\Omega^k_{Z,c}$ the \\emph{sheaf of holomorphic $k$-forms on $\\comp Z$ with compact support in $Z$} defined on connected open sets by\n\\[ \\Omega^k_{Z,c}(U):=\\begin{cases}\n\\Omega^k_Z(U) & \\text{if $U\\subseteq Z$} \\\\\n0             & \\text{otherwise.}\n\\end{cases} \\]\n\n\\begin{remark} \\label{rem:direct_image} To justify the name of the sheaf $\\Omega^k_{Z,c}$, denote by $\\iota\\colon Z\\hookrightarrow\\comp Z$ the inclusion. Then we have\n\\[ \\Omega^k_{Z,c}=\\iota_!\\Omega^k_Z. \\]\nIn other words, this is the \\emph{direct image with compact support} of the sheaf $\\Omega^k_Z$. In particular, in the case we study here, the cohomology with compact support of $\\Omega^k_Z$ is computed by the usual sheaf cohomology of $\\Omega^k_{Z,c}$, i.e., we have\n\\[ H_c^\\bul(\\comp Z, \\Omega^k_Z)=H^\\bul(\\comp Z, \\Omega^k_{Z,c}). \\qedhere \\]\n\\end{remark}\n\nWe now specify the above set-up for $Z = \\Sigma$ and $\\comp \\Sigma$ the canonical compactification of $\\Sigma$.\n\n\\medskip\n\nFor a cone $\\sigma \\in \\Sigma$, the canonical compactification $\\comp\\Sigma^\\sigma$ naturally leaves at infinity of $\\comp \\Sigma$. Using the notations of Section~\\ref{sec:tropvar}, the fan $\\Sigma_\\infty^\\sigma$, which is based at the point $\\infty_\\sigma$ of $\\comp \\Sigma$, is naturally isomorphic to $\\Sigma^\\sigma$. Via this isomorphism, the closure $\\comp{\\Sigma_\\infty^\\sigma}$ of ${\\Sigma_\\infty^\\sigma}$ in $\\comp \\Sigma$ coincides with the canonical compactification $\\comp \\Sigma^\\sigma$ of $\\Sigma^\\sigma$. In the remaining of this section, to simplify the notations, we identify $\\comp \\Sigma^\\sigma$ as $\\comp{\\Sigma_\\infty^\\sigma}$ living in $\\comp \\Sigma$.\n\n\\medskip\n\nTo each $\\sigma \\in \\Sigma$ corresponds the sheaf $\\Omega^k_{\\comp \\Sigma^\\sigma}$ of holomorphic $k$-forms on $\\comp \\Sigma^\\sigma$ which by extension by zero leads to a sheaf on $\\comp \\Sigma$. We denote this sheaf by $\\Omega^k_\\sigma$. The following proposition describes the cohomology of these sheaves.\n\n\\begin{prop} \\label{prop:cohomology_Omega_fan} Notations as above, for each pair of non-negative integers $m,k$,\nwe have\n\\[ H^m(\\comp \\Sigma, \\Omega^k_\\sigma) = \\begin{cases} H^{k,k}(\\sigma) = H^{2k}(\\sigma) & \\textrm{if $m = k$} \\\\\n0  & \\textrm{otherwise}.\n\\end{cases}\\]\n\\end{prop}\n\\begin{proof}\nWe have\n\\[H^m(\\comp \\Sigma, \\Omega^k_\\sigma) \\simeq H^m(\\comp \\Sigma^\\sigma, \\Omega^k_{\\comp \\Sigma^\\sigma}) \\simeq H^{k,m}_{\\textrm{Dolb}}(\\comp\\Sigma^\\sigma) \\simeq H^{k,m}_{\\trop}(\\comp\\Sigma^\\sigma),\\]\nby the comparison Theorem~\\ref{thm:comparison}, proved in~\\cite{JSS}, and the result follows from Theorem~\\ref{thm:HI}, proved in~\\cite{AP}.\n\\end{proof}\n\nFor a pair of faces $\\tau\\prec\\sigma$ in $\\Sigma$, we get natural inclusion maps $\\comp \\Sigma^\\sigma \\hookrightarrow \\comp\\Sigma^\\tau \\hookrightarrow \\comp \\Sigma$, leading to natural restriction maps of sheaves $\\i^*_{\\tau\\prec\\sigma}\\colon\\Omega^k_\\tau \\to\\Omega^k_\\sigma$ on $\\comp \\Sigma$. Here the map $\\i_{\\tau\\prec\\sigma}$ ($= \\i_{\\sigma \\succ\\tau}$, using our convention from Section~\\ref{sec:intro}) denotes the inclusion $\\comp\\Sigma^\\sigma\\hookrightarrow\\comp\\Sigma^\\tau$.\n\n\\medskip\n\nWe consider now the following complex of sheaves on $\\comp \\Sigma$:\n\\begin{equation}\n\\Omega^k_\\bul\\colon \\qquad \\Omega_\\conezero^k \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\ \\dims{\\sigma} =1} \\Omega^k_{\\sigma} \\to \\dots \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\ \\dims{\\sigma} =d-k} \\Omega^k_{\\sigma}\n\\end{equation}\nconcentrated in degrees $0, 1, \\dots, d-k$, given by the dimension of the cones $\\sigma$ in $\\Sigma$, whose boundary maps are given by\n\\[ \\alpha\\in\\Omega_\\sigma^k \\mapsto \\d\\alpha = \\sum_{\\zeta\\ssupface\\sigma}\\sign(\\sigma,\\zeta)\\i^*_{\\sigma\\ssubface\\zeta}(\\alpha). \\]\nWe will derive Theorem~\\ref{thm:deligne} by looking at the hypercohomology groups $\\hyp^\\bul(\\comp\\Sigma, \\Omega^k_\\bul)$ of this complex and by using the following proposition.\n\n\\begin{prop} \\label{prop:exactness_Omega}\nThe following sequence of sheaves is exact\n\\[ 0 \\to \\Omega_{\\Sigma, c}^k \\to \\Omega_\\conezero^k \\to \\bigoplus_{\\varrho \\in \\Sigma \\\\ \\dims{\\varrho}=1}\\Omega^k_\\varrho \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\ \\dims{\\sigma}=2}\\Omega^k_\\sigma \\to \\dots \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\ \\dims{\\sigma}=d-k}\\Omega^k_\\sigma \\to 0. \\]\n\\end{prop}\n\n\\begin{proof} It will be enough to prove that taking sections over nice open sets $U$ give exact sequences of $\\mathbb{R}$-vector spaces.\n\nIf $U$ is included in $\\Sigma$, clearly by definition we have $\\Omega^k_{\\Sigma,c}(U)\\simeq\\Omega^k_{\\conezero}(U)$, and the other sheaves of the sequence have no nontrivial section over $U$. Thus, the sequence is exact over $U$.\n\nIt remains to prove that, for every other nice open set $U$ intersecting $\\comp\\Sigma\\setminus\\Sigma$, the sequence\n\\[ 0 \\to \\Omega_\\conezero^k(U) \\to \\bigoplus_{\\varrho \\in \\Sigma \\\\ \\dims{\\varrho}=1}\\Omega^k_\\varrho(U) \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\ \\dims{\\sigma}=2}\\Omega^k_\\sigma(U) \\to \\dots \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\ \\dims{\\sigma}=d-k}\\Omega^k_\\sigma(U) \\to 0 \\]\nis exact. Let $\\gamma\\in\\comp\\Sigma$ be the minimum face of $U$. Let $\\sigma\\in\\Sigma$ be the sedentarity of $\\gamma$. The closed strata of $\\comp \\Sigma$ which intersect $U$ are exactly those of the form $\\comp{\\Sigma_\\infty^\\tau} \\simeq \\comp\\Sigma^{\\tau}$ with $\\tau\\prec\\sigma$. Moreover, if $\\tau$ is a face of $\\sigma$,\n\\[ \\Omega^k_\\tau(U)=\\SF^k(\\gamma). \\]\nThus, the previous sequence can be rewritten in the form\n\\[ 0 \\to \\SF^k(\\gamma) \\to \\bigoplus_{\\sigma\\prec\\sigma \\\\ \\dims{\\sigma}=1}\\SF^k(\\gamma) \\to \\bigoplus_{\\tau\\prec\\sigma \\\\ \\dims{\\tau}=2}\\SF^k(\\gamma) \\to \\dots \\to \\bigoplus_{\\tau\\prec\\sigma \\\\ \\dims{\\tau}=\\dims{\\sigma}}\\SF^k(\\gamma) \\to 0. \\]\nThis is just the cochain complex of the simplicial cohomology (for the natural simplicial structure induced by the faces) of the cone $\\sigma$ with coefficients in the group $\\SF^k(\\gamma)=\\SF_k(\\gamma)^\\dual$. This itself corresponds to the reduced simplicial cohomology of a simplex shifted by $1$. This last cohomology is trivial, thus the sequence is exact. That concludes the proof of the proposition.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:deligne}]\nBy~\\cite{JSS}, we have\n\\[ H^m(\\comp\\Sigma,\\Omega^k_{\\Sigma,c}) = H^m_c(\\Sigma,\\Omega^k_\\Sigma) = H^{k,m}_{\\trop,c}(\\Sigma)=\n\\begin{cases} \\SF^{d-k}(\\conezero)^\\dual = \\SF_{d-k}(\\conezero) & \\text{if $m=d$}\\\\\n0 & \\text{otherwise}.\n\\end{cases} \\]\nBy proposition \\ref{prop:exactness_Omega}, the cohomology of $\\Omega^k_{\\Sigma,c}$ becomes isomorphic to the hypercohomology $\\hyp(\\Sigma, \\Omega^k_\\bul)$. Thus, we get\n\\[ \\hyp(\\Sigma, \\Omega^k_\\bul)\\simeq \\SF_{d-k}(\\conezero)[d], \\]\nmeaning\n\\[\\hyp^m(\\comp\\Sigma, \\Omega^k_\\bul) = \\begin{cases}   \\SF_{d-k}(\\conezero) & \\textrm{ for $m =d$}\\\\\n0 & \\textrm{ otherwise.}\n\\end{cases}\\]\n\nOn the other hand, using the hypercohomology spectral sequence, combined with Proposition \\ref{prop:cohomology_Omega_fan}, we infer that the hypercohomology of $\\Omega^k_\\bul$ is given by the cohomology of the following complex:\n\\[ 0 \\rightarrow H^{2k}(\\conezero)[k] \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =1} H^{2k}(\\sigma)[k+1] \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =2} H^{2k}(\\sigma)[k+2] \\to \\dots \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =d-k} H^{2k}(\\sigma)[d] \\to 0. \\]\nWe thus conclude the exactness of the sequence\n\\begin{equation}\n0 \\rightarrow H^{2k}(\\conezero) \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =1} H^{2k}(\\sigma) \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =2} H^{2k}(\\sigma) \\to \\dots \\to \\bigoplus_{\\sigma \\in \\Sigma \\\\\n\\dims{\\sigma} =d-k} H^{2k}(\\sigma) \\to \\SF_{d-k}(\\conezero) \\to 0,\n\\end{equation}\nand the theorem follows.\n\\end{proof}\n\n\n\n\\subsection{Weight filtration on $\\SF^p$}\n\nIn this section, we introduce a natural filtration on the structure sheaf $\\SF^p$. This filtration could be regarded as the analog in polyhedral geometry of the weight filtration on the sheaf $\\Omega^p(\\log \\mathfrak X_0)$ of logarithmic differentials around the special fiber $\\mathfrak X_0$ of a semistable degeneration $\\mathfrak X$ over the unit disk.\n\n\\medskip\n\nLet $\\delta$ be a face of the triangulation $X$ of the tropical variety $\\mathfrak X$. First we define a filtration denoted by $\\~ W_\\bul$ on $\\SF^p(\\delta)$, and then define the weight filtration $W_\\bul$ by shifting the filtration induced by $\\~ W_\\bul$.\n\n\\medskip\n\nThe tangent space $\\TT\\delta$ of $\\delta$ is naturally included in $\\SF_1(\\delta)$. More generally, for any integer $s$, we have an inclusion $\\bigwedge^s\\TT\\delta \\subseteq \\SF_s(\\delta)$, which in turn gives an inclusion\n\\[\\bigwedge^s\\TT\\delta  \\wedge \\SF_{p-s}(\\delta) \\subseteq \\SF_p(\\delta)\n\\]\nfor any non-negative integer $s$.\n\n\\begin{defi} For any integer $s\\geq 0$, define $\\~ W_s \\SF^p(\\delta)$ as follows\n\\begin{align*}\n\\~W_s \\SF^p(\\delta) &:= \\\\\n  & \\hspace{-2em}\\Bigl\\{ \\, \\alpha \\in \\SF^p(\\delta) \\st \\textrm{restriction of $\\alpha$ on subspace } \\bigwedge^{s+1}\\TT\\delta  \\wedge \\SF_{p-s-1}(\\delta) \\textrm{ of } \\SF_p(\\delta) \\textrm{ is trivial}\\, \\Bigr\\}. \\qedhere\n\\end{align*}\n\\end{defi}\n\nRecall that for a pair of faces $\\gamma \\ssubface \\delta$, we denote by $\\i^*_{\\gamma\\ssubface\\delta}$ the restriction map $\\SF^p(\\gamma) \\to \\SF^p(\\delta)$.\n\n\\begin{prop} Let $\\gamma$ be a face of codimension one of $\\delta$. We have the following two cases.\n\\begin{itemize}\n\\item If $\\gamma$ and $\\delta$ have the same sedentarity, then the natural map $\\i^*_{\\gamma\\ssubface\\delta}\\colon\\SF^p(\\gamma) \\to \\SF^p(\\delta)$ sends $\\~ W_{s-1}\\SF^p(\\gamma)$ into $\\~ W_s\\SF^p(\\delta)$.\n\\item If $\\gamma$ and $\\delta$ do not have the same sedentarity, then the natural map $\\pi^*_{\\gamma\\ssubface\\delta}\\colon\\SF^p(\\gamma) \\to \\SF^p(\\delta)$ sends $\\~ W_s\\SF^p(\\gamma)$ into $\\~ W_s\\SF^p(\\delta)$.\n\\end{itemize}\n\\end{prop}\n\\begin{proof}\nFor the first point, let $\\nvect_{\\delta\/\\gamma}$ be a primitive normal vector to $\\TT\\gamma$ in $\\TT\\delta$. The map $\\SF_p(\\delta) \\to \\SF_p(\\gamma)$ restricted to the subspace $\\bigwedge^{s+1} \\TT\\delta  \\wedge \\SF_{p-s-1}(\\delta)$ can be decomposed as the composition of the following maps\n\\begin{align*}\n\\bigwedge^{s+1} \\TT\\delta & \\wedge \\SF_{p-s-1}(\\delta) = \\bigwedge^{s} \\TT\\gamma \\wedge \\nvect_{\\delta\/\\gamma} \\wedge \\SF_{p-s-1}(\\delta) + \\bigwedge^{s+1} \\TT\\gamma \\wedge \\SF_{p-s-1}(\\delta) \\\\\n&\\longrightarrow \\bigwedge^{s} \\TT\\gamma \\wedge \\nvect_{\\delta\/\\gamma} \\wedge \\SF_{p-s-1}(\\gamma) + \\bigwedge^{s+1} \\TT\\gamma \\wedge \\SF_{p-s-1}(\\gamma)\\lhook\\joinrel\\longrightarrow \\bigwedge^{s} \\TT\\gamma  \\wedge \\SF_{p-s}(\\gamma).\n\\end{align*}\n\nAn element of $\\~W_{s-1}\\SF^p(\\gamma)$ is thus sent to an element of $\\SF^p(\\delta)$ which restricts to zero on the subspace $\\bigwedge^{s+1} \\TT\\delta \\wedge \\SF_{p-s-1}(\\delta)$, in other words, to an element of $\\~ W_s\\SF^p(\\delta)$.\n\n\\medskip\n\nFor the second point, the projection $\\pi_{\\delta\\ssupface\\gamma}$ maps $\\TT\\delta$ onto $\\TT\\gamma$. Thus, it also maps $\\bigwedge^{s+1}\\TT\\delta$ onto $\\bigwedge^{s+1}\\TT\\gamma$. The image of an element of $\\~W_s\\SF^p(\\gamma)$ thus restricts to zero on $\\bigwedge^s\\TT\\delta$, i.e., it is an element of $\\~W_s\\SF^p(\\delta)$.\n\\end{proof}\n\nWe now consider the graded pieces $\\gr_{s}^{\\~ \\scaleto{W}{4pt}} \\SF^p(\\delta) := \\rquot {\\~ W_s \\SF^p(\\delta)} {\\~ W_{s-1}\\SF^p(\\delta)}$.\n\n\\medskip\n\nFor $\\delta \\in X$, we denote as before by $\\conezero^\\delta$ the zero cone of the fan $\\Sigma^\\delta$ induced by the triangulation $X$ around $\\delta$. The following proposition gives the description of the graded pieces of the filtration.\n\n\\begin{prop}\\label{prop:grading1}\nFor each face $\\delta$, we have\n\\[ \\gr_s^{\\~ \\scaleto{W}{4pt}}\\SF^p(\\delta) \\simeq \\bigwedge^s \\TT^\\dual\\delta \\otimes \\SF^{p-s}(\\conezero^\\delta). \\]\nHere $\\TT^\\dual\\delta$ is the cotangent space of the face $\\delta$.\n\\end{prop}\n\nIn order to prove this proposition, we introduce some useful maps. Denote by $\\pi_\\delta$ the natural projection $N^{\\sed(\\delta)}_\\mathbb{R} \\to N^\\delta_\\mathbb{R}$. We choose a projection $\\p_\\delta\\colon N^{\\sed(\\delta)}_\\mathbb{R} \\to \\TT\\delta$. Both projections naturally extend to exterior algebras, and for any integer $p$, we get maps\n\\begin{align*}\n\\p_\\delta\\colon\\,\\,& \\bigwedge^p N^{\\sed(\\delta)}_\\mathbb{R} \\to \\bigwedge^p \\TT\\delta, \\\\\n\\pi_\\delta\\colon\\,\\,& \\bigwedge^p N^{\\sed(\\delta)}_\\mathbb{R} \\to \\bigwedge^p N^\\delta_\\mathbb{R}.\n\\end{align*}\nFurthermore, we get a map\n\\[ \\pi_\\delta\\colon \\SF_p(\\delta) \\to \\SF_p(\\conezero^\\delta). \\]\n\n\\medskip\n\nWe have the corresponding pullback $\\p_\\delta^*$ and $\\pi_\\delta^*$ between the corresponding dual spaces $\\SF^p(\\delta)$ and $\\SF(\\conezero^\\delta)$. Moreover, if $\\alpha\\in\\SF^p(\\delta)$ is zero on $\\TT\\delta\\wedge\\SF_{p-1}(\\delta)$, since $\\TT\\delta=\\ker(\\pi_\\delta)$, we can define a natural pushforward $\\pi_{\\delta\\,*}(\\alpha)$ of $\\alpha$ in $\\SF_p(\\conezero^\\delta)$. This leads to a map (in fact an isomorphism)\n\\[ \\pi_{\\delta\\,*}\\colon \\~W_0(\\SF^p(\\delta)) \\to \\SF^p(\\conezero^\\delta). \\]\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:grading1}]\nLet $\\alpha$ be an element of $\\~W_s\\SF^p(\\delta)$. Let $\\u\\in \\bigwedge^s\\TT\\delta$. Recall that the contraction of $\\alpha$ by $\\u$ is the multiform $\\beta:=\\alpha(\\u\\wedge {}\\cdot{})\\in \\SF^{p-s}(\\delta)$.\n\nSince $\\alpha$ is in $\\~W_s\\SF^p(\\delta)$, and $ \\u \\wedge \\TT\\delta\\wedge \\SF_{p-s-1}(\\delta) \\subseteq \\bigwedge^{s+1} \\TT\\delta\\wedge \\SF_{p-s-1}(\\delta)$, by definition of the filtration $\\~W_\\bul$, the contracted multiform $\\beta$ is zero on $\\TT\\delta\\wedge \\SF_{p-s-1}(\\delta)$. Thus, $\\beta\\in\\~W_0(\\delta)$. Hence, we get a morphism\n\\[ \\begin{array}{rrcl}\n\\Psi\\colon \\~W_s\\SF^p(\\delta) & \\longrightarrow & \\Hom\\Bigl(\\,\\bigwedge^s\\TT\\delta\\,,\\, \\SF^{p-s}(\\conezero^\\delta)\\,\\Bigr) &\\simeq \\,\\,\\bigwedge^s \\TT^\\dual\\delta \\otimes \\SF^{p-s}(\\conezero^\\delta), \\\\[.4em]\n\\alpha  & \\longmapsto &\\quad\\bigl(\\,  \\u \\longmapsto \\pi_{\\delta\\,*}(\\alpha(\\u \\wedge {}\\cdot{}))  \\, \\bigr).\n\\end{array} \\]\nNotice that the kernel of $\\Psi$ is $\\~W_{s-1}(\\SF^p(\\delta))$, i.e., its cokernel is $\\gr^{\\~W}_s(\\SF^p(\\delta))$.\n\nSet\n\\[ \\begin{array}{rccl}\n\\Phi\\colon& \\bigwedge^s \\TT^\\dual\\delta \\otimes \\SF^{p-s}(\\conezero^\\delta), & \\longrightarrow & \\SF^p(\\delta) \\\\[.4em]\n& x \\otimes y  & \\longmapsto & \\p_\\delta^*(x)\\wedge\\pi_\\delta^*(y),\n\\end{array} \\]\nand extended by linearity.\n\n\\medskip\n\nOne verifies directly that $\\Im(\\Psi)\\subseteq\\~W_s\\SF^p(\\delta)$, and $\\Psi\\circ\\Phi$ is identity on $\\bigwedge^s \\TT^\\dual\\delta \\otimes \\SF^{p-s}(\\conezero^\\delta)$. In particular, $\\Psi$ is surjective. We thus infer that $\\Psi$ induces an isomorphism between its cokernel and $\\bigwedge^s \\TT^\\dual\\delta \\otimes \\SF^{p-s}(\\conezero^\\delta)$. This concludes the proof as we have seen that its cokernel of $\\Psi$ is $\\gr^{\\~W}_s(\\SF^p(\\delta))$.\n\\end{proof}\n\n\\medskip\n\n\\begin{defi}[Weight filtration]\nFor each $\\delta \\in X$, we define the \\emph{weight filtration} $W_\\bul$ on $\\SF^p(\\delta)$ and its \\emph{opposite filtration} $W^\\bul$ by\n\\[ W_s \\SF^p(\\delta) = \\~W_{s+\\dims\\delta}\\SF^p(\\delta), \\textrm{ and }\\]\n\\[W^s \\SF^p(\\delta) = \\~ W_{-s+\\dims{\\delta}}\\SF^p(\\delta),\\]\nfor each integer $s$.\n\\end{defi}\n\nProposition~\\ref{prop:grading1} directly translates into the following facts about the filtration $W^\\bul$.\n\n\\begin{cor} \\label{cor:grading} The following holds.\n\\begin{enumerate}\n\\item For each face $\\delta$, we have\n\\[ \\gr^s_{\\scaleto{W}{4pt}} \\SF^p(\\delta) \\simeq \\bigwedge^{\\dims{\\delta}-s} \\TT^\\dual\\delta \\otimes \\SF^{p+s-\\dims{\\delta}}(\\conezero^\\delta).\\]\n\\item For inclusion of faces $\\gamma\\prec\\delta$, the map $\\SF^p(\\gamma) \\to \\SF^p(\\delta)$ respects the filtration $W^\\bul$. In particular, we get an application at each graded piece $\\gr^s_{\\scaleto{W}{4pt}}\\SF^p(\\gamma) \\to \\gr^s_{\\scaleto{W}{4pt}} \\SF^p(\\delta)$.\n\\end{enumerate}\n\\end{cor}\n\n\n\\subsubsection{Description of the restriction maps on graded pieces of the weight filtration}\nWe would now like to explicitly describe the map induced by $\\i^*$ on the level of graded pieces. In what follows, we identify $\\gr^s_{\\scaleto{W}{4pt}} \\SF^p(\\delta)$ with the decomposition $\\bigwedge^{\\dims{\\delta}-s} \\TT^\\dual\\delta \\otimes \\SF^{p+s-\\dims{\\delta}}(\\conezero^\\delta)$ and also, by duality, with $\\Hom\\Bigl(\\,\\bigwedge^{\\dims\\delta-s}\\TT\\delta\\,,\\, \\SF^{p+s-\\dims\\delta}(\\conezero^\\delta)\\,\\Bigr)$. We call $\\bigwedge^{\\dims{\\delta}-s} \\TT^\\dual\\delta$ and $\\SF^{p+s-\\dims{\\delta}}(\\conezero^\\delta)$ the \\emph{parallel part} and the \\emph{transversal part} of the graded piece of the filtration, respectively. We use the notation $\\alpha_\\parr, \\beta_\\parr$, etc. when referring to the elements of the parallel part, and use $\\alpha_\\perp, \\beta_\\perp$, etc. when referring to the elements of the transversal part. In particular, each element of the graded piece is a sum of elements of the form $\\alpha_\\parr\\otimes \\alpha_\\perp$.\n\n\\medskip\n\nWe denote by $\\Psi_{s,\\delta}\\colon W_s\\SF^p(\\delta)\\to\\gr^s_{\\scaleto{W}{4pt}}\\SF^p(\\delta)$ the projection. As we have seen in the proof of Proposition \\ref{prop:grading1}, this map is explicitly described by\n\\[ \\begin{array}{rrl}\n\\Psi_{s,\\delta}\\colon W^s\\SF^p(\\delta) & \\longrightarrow & \\gr_{\\scaleto{W}{4pt}}^s\\SF^p(\\delta) \\\\[.4em]\n\\alpha  & \\longmapsto & \\bigl(\\,  \\u \\longmapsto \\pi_{\\delta\\,*}(\\alpha(\\u \\wedge {}\\cdot{}))  \\, \\bigr).\n\\end{array} \\]\nWe also have a section $\\Phi_{s,\\delta}$, which this time depends on the chosen projection $\\p_\\delta$, and which is given by\n\\[ \\begin{array}{rrcl}\n\\Phi_{s,\\delta}\\colon& \\gr_{\\scaleto{W}{4pt}}^s\\SF^p(\\delta) & \\longrightarrow & W^s\\SF^p(\\delta) \\\\[.4em]\n& \\alpha_\\parr \\otimes \\alpha_\\perp  & \\longmapsto & \\p_\\delta^*(\\alpha_\\parr)\\wedge\\pi_\\delta^*(\\alpha_\\perp).\n\\end{array} \\]\n\n\\medskip\n\nConsider now two faces $\\gamma\\ssubface\\delta$ in $X$. We define two maps $\\i^*_\\parr$ and $\\i^*_\\perp$ between parallel and transversal parts of the graded pieces as follows. For each non-negative integer $t$, the map\n\\[\\i^*_\\parr \\colon \\bigwedge^{t} \\TT^\\dual\\gamma \\longrightarrow   \\bigwedge^{t+1} \\TT^\\dual\\delta \\]\nsends an element $\\alpha_\\parr \\in  \\bigwedge^{t} \\TT^\\dual\\gamma$ to the element $\\beta_\\parr =  \\i^*_\\parr(\\alpha_\\parr)$ of $\\bigwedge^{t+1} \\TT^\\dual\\delta$ defined as follows. The multiform $\\beta_\\parr$ is the unique element of $\\bigwedge^{t+1} \\TT^\\dual\\delta$ which restricts to zero on $\\bigwedge^{t+1}\\TT\\gamma$ under the inclusion map $\\bigwedge^{t+1} \\TT \\gamma \\hookrightarrow \\bigwedge^{t+1} \\TT\\delta$, and which verifies\n\\[ \\beta_\\parr(\\u\\wedge \\nvect_{\\delta\/\\gamma})=\\alpha_\\parr(\\u) \\quad\\text{for any $\\u\\in\\bigwedge^{t}\\TT\\gamma\\subset\\bigwedge^{t}\\TT\\delta$}. \\]\nHere $\\nvect_{\\delta\/\\gamma}$ is any primitive normal vector to $\\TT\\gamma$ in $\\TT\\delta$. In other words,\n\\[ \\beta_\\parr=\\p_\\gamma^*(\\alpha_\\parr)\\rest{\\TT\\delta} \\wedge \\nvect_{\\delta\/\\gamma}^\\dual, \\]\nwhere $\\nvect_{\\delta\/\\gamma}^\\dual$ is the form on $\\TT\\delta$ which vanishes on $\\TT\\gamma$ and which takes value 1 on $\\nvect_{\\delta\/\\gamma}$.\n\nFor any positive integer $t$, the other map\n\\[\\i^*_\\perp\\colon \\SF^{t}(\\conezero^\\gamma) \\longrightarrow \\SF^{t-1}(\\conezero^\\delta)\\]\nbetween transversal parts is similarly defined as follows. This is the unique linear map which sends an element $\\alpha_\\perp$ of $\\SF^{t}(\\conezero^\\gamma)$ to the element $\\beta_\\perp = \\i^*_\\perp(\\alpha_\\perp)$ which verifies\n\\[ \\alpha_\\perp(\\e_{\\delta\/\\gamma}\\wedge\\v)=\\beta_\\perp(\\pi_{\\gamma\\prec\\delta}(\\v)) \\quad\\text{for any $\\v\\in \\SF_{t-1}(\\conezero^\\gamma)$}. \\]\nHere, as in the previous sections, $\\e_{\\delta\/\\gamma}$ is the primitive vector of the ray $\\rho_{\\delta\/\\gamma}$ corresponding to $\\delta$ in $\\Sigma^\\gamma$, and $\\pi_{\\gamma\\prec\\delta}$ is the projection from $\\comp\\Sigma^\\gamma$ to $\\comp\\Sigma^\\delta$ (along $\\mathbb{R} \\e_{\\delta\/\\gamma}$), which naturally induces a surjective map from $\\SF_{t-1}(\\conezero^\\gamma)\\to \\SF_{t-1}(\\conezero^\\delta)$. In other words, $\\beta_\\perp$ is the pushforward by $\\pi_{\\gamma\\prec\\delta}$ of the contraction of $\\alpha_\\perp$ by $\\e_{\\delta\/\\gamma}$ (which is well-defined because $\\ker{\\pi_{\\gamma\\prec\\delta}}=\\mathbb{R} \\e_{\\delta\/\\gamma}$).\n\n\\begin{prop}\\label{prop:grading} Let $\\gamma\\ssubface\\delta$ be two faces in $X$. Notations as above, the induced map on graded pieces $\\gr^s_{\\scaleto{W}{4pt}} \\SF^p(\\gamma) \\to \\gr^s_{\\scaleto{W}{4pt}} \\SF^p(\\delta)$ is zero provided that $\\sed(\\gamma)\\neq\\sed(\\delta)$. Otherwise, it coincides with the map\n\\[ \\begin{array}{rrclcrcl}\ni^*_\\parr \\otimes \\i^*_\\perp\\colon & \\bigwedge^{\\dims{\\gamma}-s} \\TT^\\dual\\gamma &\\otimes& \\SF^{p+s-\\dims{\\gamma}}(\\conezero^\\gamma) & \\longrightarrow & \\bigwedge^{\\dims{\\delta}-s} \\TT^\\dual\\delta &\\otimes& \\SF^{p+s - \\dims{\\delta}}(\\conezero^\\delta), \\\\\n& \\alpha_\\parr &\\otimes& \\alpha_\\perp & \\longmapsto & \\sign(\\gamma,\\delta) \\, \\beta_\\parr &\\otimes& \\beta_\\perp,\n\\end{array} \\]\nwhere $\\beta_\\parr = \\i^*_\\parr(\\alpha_\\parr)$ and $\\beta_\\perp =\\i^*_\\perp(\\alpha_\\perp)$.\n\\end{prop}\n\n\\begin{proof}\nIf $\\sed(\\delta)\\neq\\sed(\\gamma)$, then the natural map $\\SF^p(\\gamma)\\to\\SF^p(\\delta)$ is $\\pi_{\\gamma\\ssubface\\delta}^*$, and it sends $W^s\\SF^p(\\gamma)$ onto $W^{s+1}\\SF^p(\\delta)$. Thus, it induces the zero map on the graded pieces.\n\n\\medskip\n\nAssume $\\sed(\\delta)=\\sed(\\gamma)$. Let $\\alpha=\\alpha_\\parr\\otimes\\alpha_\\perp\\in \\gr_{\\scaleto{W}{4pt}}^s\\SF^p(\\gamma)$. By definition, the map induced by $\\i^*$ on the graded pieces maps $\\alpha$ onto $\\Psi_{s+1,\\delta}\\circ\\i^*_{\\gamma\\ssubface\\delta}\\circ\\Phi_{s,\\gamma}(\\alpha)$. To understand this image, take an element $\\u_\\parr\\otimes\\u_\\perp$ in $\\bigwedge^{\\dims\\delta-s}\\TT\\delta\\otimes \\SF_{p+s-\\dims\\delta}(\\conezero^\\delta)$. By definition of $\\Psi_{s+1,\\delta}$ we get\n\\[ \\Psi_{s+1,\\delta}\\circ \\i^*_{\\gamma\\ssubface\\delta}\\circ \\Phi_{s,\\gamma}(\\alpha)\\,\\bigl(\\u_\\parr\\otimes\\u_\\perp\\bigr) = \\i^*_{\\gamma\\ssubface\\delta}\\circ \\Phi_{s,\\gamma}(\\alpha)\\,\\bigl(\\u_\\parr\\wedge\\~\\u_\\perp\\bigr) , \\]\nwhere $\\~\\u_\\perp$ is any preimage of $\\u_\\perp$ by $\\Phi_{s,\\gamma}$. Then,\n\\begin{align*}\n\\i^*_{\\gamma\\ssubface\\delta}\\circ \\Phi_{s,\\gamma}(\\alpha)\\,\\bigl(\\u_\\parr\\wedge\\~\\u_\\perp\\bigr)\n  &= \\Phi_{s,\\gamma}(\\alpha)\\,\\bigl(\\u_\\parr\\wedge\\~\\u_\\perp\\bigr) \\\\\n  &= \\bigl(\\p^*_\\gamma(\\alpha_\\parr)\\wedge\\pi^*_\\gamma(\\alpha_\\perp)\\bigr)\\bigl(\\u_\\parr\\wedge\\~\\u_\\perp\\bigr).\n\\end{align*}\nHere, $\\u_\\parr \\in \\bigwedge^{\\dims\\delta-s} \\TT\\delta$, and we have the decomposition\n\\[ \\bigwedge^{\\dims\\delta-s} \\TT\\delta = \\bigwedge^{\\dims\\gamma-s+1}\\TT \\gamma \\oplus \\bigwedge^{\\dims\\gamma-s} \\TT\\gamma \\wedge\\nvect_{\\delta, \\gamma}, \\]\nwhere $\\nvect_{\\delta\/\\gamma}$ is a primitive normal vector to $\\TT\\gamma$ in $\\TT\\delta$.\nThus, it suffices to study the following two cases.\n\\begin{itemize}\n\\item Assume that $\\u_\\parr=\\u'_\\parr\\wedge\\nvect_{\\delta\/\\gamma}$ with $\\u'_\\parr\\in\\bigwedge^{\\dims\\gamma-s}\\TT\\gamma$. Since $\\TT\\gamma=\\ker(\\pi_\\gamma)$, we get\n\\begin{align*}\n\\bigl(\\p^*_\\gamma(\\alpha_\\parr)\\wedge\\pi^*_\\gamma(\\alpha_\\perp)\\bigr)\\bigl(\\u'_\\parr\\wedge\\nvect_{\\delta\/\\gamma}\\wedge\\~\\u_\\perp\\bigr)\n  &= \\alpha_\\parr(\\p_\\gamma(\\u_\\parr)) \\cdot \\alpha_\\perp(\\pi_\\gamma(\\nvect_{\\delta\/\\gamma} \\wedge \\~\\u_\\perp)) \\\\\n  &= \\alpha_\\parr(\\u'_\\parr) \\cdot \\alpha_\\perp(\\e_{\\delta\/\\gamma} \\wedge \\pi_\\gamma(\\~\\u_\\perp)) \\\\\n  &= \\beta_\\parr(\\u_\\parr)\\cdot\\beta_\\perp(\\u_\\perp),\n\\end{align*}\nsince $\\u_\\parr=\\u'_\\parr\\wedge\\nvect_{\\delta\/\\gamma}$ and $\\u_\\perp=\\pi_{\\gamma\\prec\\delta}(\\pi_\\gamma(\\~\\u_\\perp))$.\n\\item If $\\u_\\parr\\in\\bigwedge^{\\dims\\gamma-s+1} \\TT\\gamma$, since $\\p_\\gamma^*(\\alpha_\\parr)\\in\\bigwedge^{\\dims\\gamma-s} N^{\\sed(\\gamma)}_\\mathbb{R}$, a part of $\\u_\\parr$ must be evaluated by $\\pi_\\gamma^*(\\alpha_\\perp)$. But this evaluation will be zero since $\\TT_\\gamma=\\ker(\\pi_\\gamma)$. Thus,\n\\begin{align*}\n0\n  &= \\bigl(\\p^*_\\gamma(\\alpha_\\parr)\\wedge\\pi^*_\\gamma(\\alpha_\\perp)\\bigr)(\\u'_\\parr\\wedge\\nvect_{\\delta\/\\gamma}\\wedge\\~\\u_\\perp) \\\\\n  &= \\beta_\\parr(\\u_\\parr)\\cdot\\beta_\\perp(\\u_\\perp).\n\\end{align*}\n\\end{itemize}\nIn any case, the statement of the proposition holds.\n\\end{proof}\n\nUsing the above proposition, we identify in the sequel the graded pieces of the opposite weight filtration and the maps between them by $\\bigwedge^{\\bul} \\TT^\\dual\\gamma \\otimes \\SF^{\\bul}(\\conezero^\\gamma)$ and maps between them.\n\n\n\n\\subsection{Yoga of spectral sequences and proof of the main theorem} \\label{sec:steenbrinkdoublecomplex}\n\nIn this section, we study the combinatorics behind the Steenbrink spectral sequence, and use this to prove Theorem~\\ref{thm:steenbrink}.\nWe introduce two spectral sequences. One of these spectral sequences computes the tropical cohomology $H_\\trop^{p,\\bul}$, and the another one computes the cohomology of the $2p$-th row of the tropical Steenbrink spectral sequence. We show that the two spectral sequences are isomorphic in page one. Then using a \\emph{spectral resolution lemma} we generalize the isomorphism in page one to all pages, from which we will deduce that these two cohomologies coincide as stated by Theorem \\ref{thm:steenbrink}.\n\n\\medskip\n\nConsider first the cochain complex $C^\\bul(X, \\SF^p)$ which calculates the tropical cohomology groups $H_\\trop^{p,q}(X)$. By Corollary~\\ref{cor:grading}, the decreasing filtration $W^\\bul$ on $\\SF^p$ induces a decreasing filtration on the cochain complex $C^\\bul(X, \\SF^p)$. By an abuse of the notation, we denote this filtration by $W^\\bul$. This leads to a spectral sequence\n\\begin{equation} \\label{eqn:abutment_trop}\n\\CCp{p}_0^{\\bul,\\bul} \\Longrightarrow H^\\bul(X, \\SF^p) = H^{p,\\bul}_{\\trop}(X)\n\\end{equation}\nwhere\n\\[ \\CCp{p}_0^{a,b} = \\gr_{\\scaleto{W}{4pt}}^a C^{a+b}(X, \\SF^p) = \\bigoplus_{\\delta\\in X \\\\ \\dims{\\delta}=a+b}\\bigwedge^b\\TT^\\dual\\delta\\otimes\\SF^{p-b}(\\conezero^\\delta) \\]\nand the differentials in page zero, which are of bidegree $(0,1)$, are given by Proposition~\\ref{prop:grading}. We call this the \\emph{tropical spectral sequence}. The zero-th page of this spectral sequence is given in Figure \\ref{fig:tropical_spectral_sequence}. The dashed arrows correspond to the maps of the first page. The explicit form of all these maps appear later in this section.\n\n\\begin{figure}\n\\caption{The zero-th page of the tropical spectral sequence $\\CCp{p}_0^{\\bul,\\bul}$ over $\\SF^p$.} \\label{fig:tropical_spectral_sequence}\n\\renewcommand\\S\\CCab\n\\newcommand{\\hspace{-2ex}\\cddots\\hspace{-2ex} }{\\hspace{-2ex}\\raisebox{2pt}[8pt][4pt]{\\makebox[20pt]{$\\dots$}}\\hspace{-2ex} }\n\\small\n\\[ \\begin{tikzcd}\n\\S00p      \\dar[\"\\i^*_\\parr\\otimes\\i^*_\\perp\"]\\rar[dashed, \"\\pi^*+\\d^\\i\"]&  \\S10p      \\dar&  \\hspace{-2ex}\\cddots\\hspace{-2ex}   &  \\S{d-p}0p        \\dar\\\\\n\\S11{p-1}                                                 \\dar\\rar[dashed]&  \\S21{p-1}  \\dar&  \\hspace{-2ex}\\cddots\\hspace{-2ex}   &  \\S{d-p+1}1{p-1}  \\dar\\\\\n\\raisebox{0pt}[14pt][4pt]{\\makebox[10pt]{$\\vdots$}}                                                                      \\dar&  \\raisebox{0pt}[14pt][4pt]{\\makebox[10pt]{$\\vdots$}}   \\dar&  \\hspace{-2ex}\\cddots\\hspace{-2ex}   &  \\raisebox{0pt}[14pt][4pt]{\\makebox[10pt]{$\\vdots$}}         \\dar\\\\\n\\S{p}p0                                                       \\rar[dashed]&  \\S{p+1}p0      &  \\hspace{-2ex}\\cddots\\hspace{-2ex}   &  \\S{d}p0\n\\end{tikzcd}\n\\]\n\\end{figure}\n\nBefore introducing the second spectral sequence, let us consider the $2p$-th row $\\ST^{\\bul,2p}_1$ of the Steenbrink spectral sequence. This row can be decomposed into a double complex as follows. We define the double complex $\\STp{p}^{\\bul, \\bul}$ by\n\\[ \\STp{p}^{a,b}:=\\begin{cases}\n  \\bigoplus_{\\delta\\in X_\\f \\\\ \\dims\\delta=p+a-b} H^{2b}(\\delta) & \\text{if $a\\geq0$ and $b\\leq p$}, \\\\\n  0 & \\text{otherwise,}\n\\end{cases} \\]\nwhose differential of bidegree $(1,0)$ is $\\i^*$ and whose differential of bidegree $(0,1)$ is $\\gys$. From the identification\n\\[ \\STp{p}^{a,b}=\\ST^{a+b-p,2p,p+a-b}, \\]\nand the following equivalence\n\\[ p+a-b\\geq\\abs{a+b-p} \\Longleftrightarrow a\\geq0 \\text{ and } b\\leq p, \\]\nwe deduce that\n\\[ \\ST^{\\bul,2p}_1=\\Tot^\\bul(\\STp{p}^{\\bul,\\bul})[-p]. \\]\nThe double complex $\\STp{p}^{\\bul,\\bul}$ is represented in Figure \\ref{fig:STp}.\n\n\\begin{figure}\n\\caption{The $p$-th unfolded Steenbrink double complex $\\STp{p}^{\\bul,\\bul}$.} \\label{fig:STp}\n\\small\n{ \\renewcommand\\S\\STpab\n\\[ \\begin{tikzcd}\n\\S{p}0     \\dar[\"\\gys\"']\\rar[\"\\i^*\"]&  \\S{p+1}0   \\dar\\rar&  \\raisebox{2pt}[8pt][4pt]{\\makebox[20pt]{$\\dots$}}   \\rar&  \\S{d}0       \\dar\\\\\n\\S{p-1}2                    \\dar\\rar&  \\S{p}2     \\dar\\rar&  \\raisebox{2pt}[8pt][4pt]{\\makebox[20pt]{$\\dots$}}   \\rar&  \\S{d-1}2     \\dar\\\\\n\\raisebox{0pt}[14pt][4pt]{\\makebox[10pt]{$\\vdots$}}                        \\dar&  \\raisebox{0pt}[14pt][4pt]{\\makebox[10pt]{$\\vdots$}}       \\dar&  \\raisebox{0pt}[14pt][4pt]{\\makebox[20pt]{$\\ddots$}}      &  \\raisebox{0pt}[14pt][4pt]{\\makebox[10pt]{$\\vdots$}}     \\dar\\\\\n\\S{0}{2p}                       \\rar&  \\S{1}{2p}      \\rar&  \\raisebox{2pt}[8pt][4pt]{\\makebox[20pt]{$\\dots$}}   \\rar&  \\S{d-p}{2p}\n\\end{tikzcd} \\] }\n\\end{figure}\n\n\\medskip\n\nFinally, we introduce another double complex $\\STinf{p}^{\\bul,\\bul}$, which will play the role of a bridge between $\\STp{p}^{\\bul,\\bul}_0$ and $\\CCp{p}^{\\bul,\\bul}_0$ via Proposition~\\ref{prop:STinf_to_ST} and Theorem~\\ref{thm:spectral_isomorphism} below. Recall that for $\\delta\\in X$, the max-sedentarity of $\\delta$ is defined by\n\\[ \\maxsed(\\delta) := \\max\\{\\sed(x) \\st x\\in\\delta\\} \\in X_\\infty. \\]\nThe double complex $\\STinf{p}^{\\bul,\\bul}$ is defined by\n\\[ \\STinf{p}^{\\bul,\\bul}=\\bigoplus_{\\sigma\\in X_\\infty}\\STinf{p}^{\\sigma,\\bul,\\bul} \\]\nwhere, for $\\sigma\\in X_\\infty$, we set\n\\[ \\STinf{p}^{\\sigma,a,b}:=\n\\begin{cases}\n  \\bigoplus_{\\delta\\in X \\\\ \\maxsed(\\delta)=\\sigma \\\\ \\dims\\delta=p+a-b \\\\ \\dims{\\delta_\\infty}\\leq a} H^{2b}(\\delta) & \\text{if $b\\leq p$}, \\\\\n  0 & \\text{otherwise.}\n\\end{cases} \\]\nOn $\\STinf{p}^{\\sigma,\\bul,\\bul}$, the differentials of bidegree $(0,1)$ are given by $\\gys$, and those of bidegree $(1,0)$ are given by $\\i^*+\\pi^*$. Note that, on the whole complex $\\STinf{p}^{\\bul,\\bul}$, no differentials goes from $\\STinf{p}^{\\sigma,\\bul,\\bul}$ to $\\STinf{p}^{\\sigma',\\bul,\\bul}$ if $\\sigma\\neq\\sigma'$. In other words, the differentials $\\gys$ and $\\i^*$ are restricted to those pairs of faces $\\gamma\\ssubface\\delta$ such that $\\maxsed(\\gamma)=\\maxsed(\\delta)$ and $\\sed(\\gamma)=\\sed(\\delta)$, or equivalently such that $\\dims{\\gamma_\\infty}=\\dims{\\delta_\\infty}$ and $\\sed(\\gamma)=\\sed(\\delta)$.\n\nThe total double complex $\\STinf{p}^{\\bul,\\bul}$ is represented in Figure \\ref{fig:STinf}.\n\n\\begin{figure}\n\\caption{The $p$-th extended Steenbrink double complex $\\STinf{p}^{\\bul,\\bul}$. The filtration by columns gives the spectral sequence $\\STinfI{p}^{\\bul,\\bul}_\\bul$. Note that here, $\\gys$ has to be restricted to pairs of faces with the same max-sedentarity.} \\label{fig:STinf}\n\\small\n{ \\renewcommand{\\S}[3]{\\STinfab{\\delta}{#1}{#3}{#2}}\n\\[ \\begin{tikzcd}\n\\S{p}00     \\dar[\"\\gys\"']\\rar[\"\\i^*+\\pi^*\"]&  \\S{p+1}01   \\dar\\rar&  \\raisebox{2pt}[8pt][4pt]{\\makebox[20pt]{$\\dots$}}   \\rar&  \\S{d}0{d-p}       \\dar\\\\\n\\S{p-1}20                          \\dar\\rar&  \\S{p}21     \\dar\\rar&  \\raisebox{2pt}[8pt][4pt]{\\makebox[20pt]{$\\dots$}}   \\rar&  \\S{d-1}2{d-p}     \\dar\\\\\n\\raisebox{0pt}[14pt][4pt]{\\makebox[10pt]{$\\vdots$}}                               \\dar&  \\raisebox{0pt}[14pt][4pt]{\\makebox[10pt]{$\\vdots$}}        \\dar&  \\raisebox{0pt}[14pt][4pt]{\\makebox[20pt]{$\\ddots$}}      &  \\raisebox{0pt}[14pt][4pt]{\\makebox[10pt]{$\\vdots$}}          \\dar\\\\\n\\S{0}{2p}0                             \\rar&  \\S{1}{2p}1      \\rar&  \\raisebox{2pt}[8pt][4pt]{\\makebox[20pt]{$\\dots$}}   \\rar&  \\S{d-p}{2p}{d-p}\n\\end{tikzcd} \\] }\n\\end{figure}\n\n\\medskip\n\nFor the cone $\\conezero$ of $X_\\infty$, the double complex $\\STinf{p}^{\\,\\conezero,\\bul,\\bul}$ is described as follows. Faces $\\delta$ such that $\\maxsed(\\delta)=\\conezero$ are exactly faces of $X_\\f$. Moreover, this implies that $\\dims{\\delta_\\infty}=0$. Thus, $\\dims{\\delta_\\infty}\\leq a$ is equivalent to $a\\geq0$. Hence,\n\\[ \\STinf{p}^{\\,\\conezero,\\bul,\\bul} = \\STp{p}^{\\bul,\\bul}. \\]\n\nFor any non-zero cone $\\sigma\\in X_\\infty$, we have the following result whose proof will be given at the end of this section.\n\\begin{prop} \\label{prop:STinf_to_ST}\nLet $\\sigma\\in X_\\infty\\setminus\\{0\\}$. Then the cohomology of $\\Tot^\\bul(\\STinf{p}^{\\sigma,\\bul,\\bul})$ is zero.\n\\end{prop}\n\n\\begin{proof}\nSection \\ref{sec:STinf_to_ST} is devoted to the proof of this proposition.\n\\end{proof}\n\nFiltration by columns of the double complex $\\STinf{p}^{\\bul,\\bul}$ gives a spectral sequence $\\STinfI{p}_0^{\\bul,\\bul}$, which abuts to the cohomology of the total complex of $\\STinf{p}^{\\bul,\\bul}$. By the previous discussion and Proposition~\\ref{prop:STinf_to_ST}, this cohomology is just the cohomology of $\\Tot^\\bul(\\STp{p}^{\\bul,\\bul})$, i.e.,\n\\begin{equation} \\label{eqn:abutment_STinf}\n\\STinfI{p}_0^{\\bul,\\bul} \\Longrightarrow H^\\bul(\\Tot^\\bul(\\STp{p}^{\\bul,\\bul})) = H^{\\bul}(\\ST^{\\bul,2p}_1)[p].\n\\end{equation}\n\n\\medskip\n\nThe link between the two spectral sequences $\\CCp{p}_\\bul^{\\bul,\\bul}$ and $\\STinfI{p}_\\bul^{\\bul,\\bul}$ is summarized by the following theorem.\n\\begin{thm} \\label{thm:spectral_isomorphism}\nThe Tropical Deligne exact sequence induces compatible canonical isomorphisms\n\\[ \\CCp{p}_k^{\\bul,\\bul}\\simeq\\STinfI{p}_k^{\\bul,\\bul}, \\]\nbetween the $k$-th pages for any $k\\geq1$.\n\\end{thm}\n\\begin{proof}\nSection \\ref{sec:CC_to_STinf} is devoted to the proof of the isomorphism between first pages, and Section \\ref{sec:every_pages} extends the isomorphism to further pages.\n\\end{proof}\n\n\\smallskip\nWe can now present the proof of the main theorem of this section.\n\\begin{proof}[Proof of Theorem \\ref{thm:steenbrink}]\nWe recall that we have to prove\n\\[ H^\\bul(\\ST^{\\bul,2p}_1, \\d)[p] = H^{p,\\bul}_{\\trop}(\\mathfrak X). \\]\n\nBy Theorem \\ref{thm:spectral_isomorphism}, we have\n\\[ \\CCp{p}_k^{\\bul,\\bul} \\simeq \\STinfI{p}_k^{\\bul,\\bul}, \\]\nfor any $k\\geq0$.\nIn particular, both spectral sequences have the same abutment. This implies y \\eqref{eqn:abutment_trop} and \\eqref{eqn:abutment_STinf} that we have a (non-canonical) isomorphism\n\\[ H^\\bul(\\ST^{\\bul,2p}_1, \\d)[p] \\simeq H^{p,\\bul}_{\\trop}(\\mathfrak X). \\]\n\\end{proof}\n\n\\begin{remark}\nWe note that though we cannot expect to have a canonical isomorphism $H^\\bul(\\ST^{\\bul,2p}_1, \\d)[p] \\simeq H^{p,\\bul}_{\\trop}(\\mathfrak X)$, the isomorphism\n\\[ \\gr_F^s H^\\bul(\\ST^{\\bul,2p}_1, \\d)[p] \\simeq \\gr_{\\scaleto{W}{4pt}}^s H^{p,\\bul}_{\\trop}(\\mathfrak X) \\]\nbetween the graduations is canonical for any $s$. Here the filtration $F^\\bul$ is the filtration induced by columns of $\\STp{p}^{\\bul,\\bul}$. The isomorphism between the two cohomology groups above should depend on the data of a smooth deformation of $\\mathfrak X$.\n\\end{remark}\n\n\\medskip\n\nIn the remaining of this section, we present the proofs of Theorem~\\ref{thm:spectral_isomorphism}, Proposition \\ref{prop:STinf_to_ST} and Proposition~\\ref{prop:app1}.\n\n\n\n\\subsection{From $\\CCp{p}_1^{\\bul,\\bul}$ to $\\STinfI{p}_1^{\\bul,\\bul}$} \\label{sec:CC_to_STinf}\n\nIn this section and the next one, we prove Theorem~\\ref{thm:spectral_isomorphism}, which states that $\\CCp{p}^{\\bul,\\bul}_k$ is canonically isomorphic to $\\STinf{p}^{\\bul,\\bul}_k$ for $k\\geq1$. As a warm-up, in this section, we study the case of the first page $k=1$. Some of the results and constructions in this section will be used in the next section to achieve the isomorphism of other pages. In order to keep the reading flow, a few technical points of this section will be treated in Appendix~\\ref{sec:technicalities}.\n\n\\medskip\n\nWe have to prove that, for every integer $a$, the cochain complex\n{ \\renewcommand{\\S}[3]{\\displaystyle \\bigoplus_{\\dims{\\delta}=#1}\\bigwedge^{#2}\\TT^\\dual\\delta\\otimes\\SF^{#3}(\\conezero^\\delta)}\n\\begin{align*}\n\\CCp{p}^{a,\\bul}_0\\colon \\qquad 0 \\to \\S{a}0{p} &\\to \\S{a+1}1{p-1} \\to \\cdots \\\\ & \\qquad\\dots\\to \\S{a+p}p0 \\to 0\n\\end{align*} }\nis quasi-isomorphic to\n{ \\renewcommand{\\S}[3]{\\bigoplus_{\\dims{\\delta}=#1 \\\\ \\dims{\\delta_\\infty}\\leq#2} H^{#3}(\\delta)}\n\\[ \\STinf{p}^{a,\\bul}_0\\colon \\qquad 0 \\to \\S{p+a}a0 \\to \\S{p+a-1}a2 \\to \\dots \\to \\S{a}a{2p} \\to 0. \\] }\nMoreover, the induced isomorphisms in cohomology must commute with the differentials of degree $(1,0)$ on the respective first pages:\n\\[ \\begin{tikzcd}\n\\CCp{p}_1^{a,\\bul} \\dar[\"\\sim\"{above,sloped}]\\rar& \\CCp{p}_1^{a+1,\\bul} \\dar[\"\\sim\"{above,sloped}]\\\\\n\\STinfI{p}_1^{a,\\bul} \\rar[\"\\i^*+\\pi^*\"]& \\STinfI{p}_1^{a+1,\\bul}\n\\end{tikzcd} \\]\n\nIn order to calculate the cohomology of the first cochain complex, we use the tropical Deligne resolution. Applying that exact sequence to the unimodular fan $\\Sigma^\\delta$ and to any integer $s$, we get the exact sequence\n\\[0\\to \\SF^s(\\conezero^\\delta) \\to \\bigoplus_{\\zeta \\in \\Sigma^{\\delta}\\\\ \\dims \\zeta =s} H^0(\\zeta) \\to \\bigoplus_{\\zeta \\in \\Sigma^{\\delta}\\\\ \\dims \\zeta =s-1} H^2(\\zeta) \\to \\dots \\to \\bigoplus_{\\zeta \\in \\Sigma^{\\delta}\\\\ \\dims \\zeta =1} H^{2s-2}(\\zeta) \\to H^{2s}(\\conezero^\\delta) \\to 0.\\]\n\nGiven the correspondence between cones $\\zeta \\in \\Sigma^\\delta$, and faces $\\eta \\in X$ which contain $\\delta$, with the same sedentarity, we rewrite the exact sequence above in the form\n{ \\renewcommand{\\S}[2]{\\bigoplus_{\\eta \\succ \\delta\\\\ \\dims\\eta=#1 \\\\ \\makebox[0pt]{\\scriptsize $\\sed(\\eta)=\\sed(\\delta)$}} H^{#2}(\\eta)}\n\\[ 0\\to \\SF^s(\\conezero^\\delta) \\to \\S{s+\\dims\\delta}0 \\to \\S{s-1+\\dims\\delta}2 \\to \\dots \\to \\S{1+\\dims\\delta}{2s-2} \\to H^{2s}(\\delta) \\to 0. \\]\n}\n\n\\medskip\n\nReplacing now each $\\SF^s(\\conezero^\\delta)$ in $\\CCp{p}_0^{a,\\bul}$ by the resolution given by the tropical Deligne complex, we get the double complex $\\Da{a}^{b, b'}$ whose total complex has the same cohomology as $\\CCp{p}^{a,\\bul}$. More precisely, define the double complex $\\Da{a}^{\\bul, \\bul}$ as follows.\n\\[ \\Da{a}^{b, b'} := \\Dab{a+b}b{p+a-b'}{2b'}. \\]\nThe differential $\\d'$ of bidegree $(0,1)$ comes from the Deligne sequence and is $\\id\\otimes\\gys$. The differential of bidegree $(1,0)$ is defined as follows, thanks to the map $\\i^*_\\parr\\colon \\bigwedge^s\\TT^\\dual\\gamma\\to\\bigwedge^{s+1}\\TT^\\dual\\delta$ which has been defined right before Proposition~\\ref{prop:grading}. The differential $\\d$ of bidegree $(1,0)$ is chosen to be $\\i^*_\\parr\\otimes\\id$ (extended using our $\\sign$ function as in Section \\ref{sec:basic_maps}) on rows with even indices and to be $-\\i^*_\\parr\\otimes\\id$ on rows with odd indices. More concisely, we set\n\\[ \\d:=(-1)^{b'}\\i^*_\\parr\\otimes\\id. \\]\n\n\\begin{prop} \\label{prop:D_commuting}\nWe have $\\d\\d'+\\d'\\d=0$. Moreover, the inclusion $(\\CCp{p}_0^{a,\\bul},\\i^*_\\parr\\otimes\\i^*_\\perp) \\hookrightarrow (\\Da{a}^{\\bul,0},\\d')$ is a morphism of cochain complexes.\n\\end{prop}\n\n\\begin{proof}\nThis follows by a simple computation. The details are given in Appendix~\\ref{sec:technicalities}.\n\\end{proof}\n\nBy the exactness of the tropical Deligne sequence that we proved in Proposition \\ref{thm:deligne}, we know that the $b$-th column of $\\Da{a}^{b,\\bul}_0$ of $\\Da{a}_0^{\\bul,\\bul}$ is a right resolution of $\\CCp{p}^{a,b}_0$.\n\n\\medskip\n\nWe claim the following result.\n\\begin{prop} \\label{prop:D_to_STinf}\nThe $b'$-th row $\\Da{a}^{\\bul,b'}_0$ of $\\Da{a}_0^{\\bul,\\bul}$ is a right resolution of $\\STinf{p}^{a,b'}_0$.\n\\end{prop}\n\nThe proof of this proposition is given in Section~\\ref{sec:D_to_STinf}. Admitting this result for the moment, we explain how to finish the proof of the isomorphism between first pages. We will need the following Lemma, which seems to be folklore, for which we provide a proof.\n\n\\begin{lemma}[Zigzag isomorphism] \\label{lem:zigzag}\nLet $\\AA^{\\bul,\\bul}$ be a double complex of differentials $\\d$ and $\\d'$ of respective degree $(1,0)$ and $(0,1)$. Assume that\n\\begin{itemize}\n\\item $\\d\\d'+\\d'\\d=0$,\n\\item $\\AA^{b,b'}=0$ if $b<0$ or $b'<0$,\n\\item $\\AA^{b,\\bul}$ is exact if $b>0$,\n\\item $\\AA^{\\bul,b'}$ is exact if $b'>0$.\n\\end{itemize}\nThen, there is a canonical isomorphism\n\\[ H^\\bul(\\AA^{0,\\bul})\\simeq H^\\bul(\\AA^{\\bul,0}). \\]\n\nMoreover, if $\\textnormal{\\textsf{B}}^{\\bul,\\bul}$ is another double complex with the same property, and if $\\Phi\\colon\\AA^{\\bul,\\bul}\\to\\textnormal{\\textsf{B}}^{\\bul,\\bul}$ is a morphism of double complexes, then the following diagram commutes.\n\\[ \\begin{tikzcd}\nH^\\bul(\\AA^{0,\\bul})  \\dar[\"\\sim\"{above, sloped}]\\rar[\"\\Phi\"]&  H^\\bul(\\textnormal{\\textsf{B}}^{0,\\bul})  \\dar[\"\\sim\"{above, sloped}] \\\\\nH^\\bul(\\AA^{\\bul,0})                             \\rar[\"\\Phi\"]&  H^\\bul(\\textnormal{\\textsf{B}}^{\\bul,0})\n\\end{tikzcd} \\]\nwhere $\\Phi$ denotes the induced maps on the cohomologies. This also holds if $\\Phi$ anticommutes with the differentials.\n\\end{lemma}\nThe isomorphism $ H^\\bul(\\AA^{0,\\bul})\\simeq H^\\bul(\\AA^{\\bul,0})$ comes from the inclusions\n\\[(\\AA^{0,\\bul}, \\d') \\hookrightarrow (\\Tot(\\AA^{\\bul,\\bul}, \\d+\\d')) \\hookleftarrow (\\AA^{\\bul,0}, \\d)\\]\nwhich are both quasi-isomorphisms. In the following, we give a canonical description of this isomorphism.\n\n\\medskip\n\n{\n\\renewcommand{\\K}{\\textnormal{\\textsf{L}}}\n\\renewcommand{\\L}{\\textnormal{\\textsf{R}}}\n\nLet $b$ be a non-negative integer and $b'$ be any integer. Define\n\\[\\K^{b,b'} := \\frac{\\ker(\\d\\d')\\cap\\AA^{b,b'}}{\\bigl(\\Im(\\d)+\\Im(\\d')\\bigr)\\cap\\AA^{b,b'}} \\qquad \\textrm{and} \\qquad \\L^{b,b'}:= \\frac{\\ker(\\d)\\cap\\ker(\\d')\\cap\\AA^{b,b'+1}}{\\Im(\\d\\d')\\cap\\AA^{b,b'+1}}. \\]\n\nWe claim the following.\n\\begin{claim} Notations as in Lemma~\\ref{lem:zigzag}, the map $\\d'$ induces an isomorphism\n\\[ \\K^{b'b'} \\underset{\\d'}{\\xrightarrow{\\ \\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$\\ }}} \\mathbb{R}^{b,b'}. \\]\n\\end{claim}\n\\begin{proof}\nLet $y$ be an element of $\\ker(\\d)\\cap\\ker(\\d')\\cap\\AA^{b,b'+1}$. By the exactness of the $b$-th column, there exists a preimage $x$ of $y$ by $\\d'$. Moreover, $x\\in\\ker(\\d\\d')\\cap\\AA^{b,b'}$. Hence, $\\d'$ is surjective from $\\ker(\\d\\d')\\cap\\AA^{b,b'}$ to $\\ker(\\d)\\cap\\ker(\\d')\\cap\\AA^{b,b'+1}$. From the trivial identity $\\Im(\\d\\d')=\\d'\\Im(\\d)$, we get\n\\[ \\d^{\\prime-1}\\bigl(\\Im(\\d\\d')\\cap\\AA^{b,b'+1}\\bigr)=\\bigl(\\Im(\\d)+\\ker(\\d')\\bigr)\\cap\\AA^{b,b'}=\\bigl(\\Im(\\d)+\\Im(\\d')\\bigr)\\cap\\AA^{b,b'}. \\]\nThis gives the isomorphism $\\d'\\colon\\K^{b,b'}\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}}\\L^{b,b'+1}$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:zigzag}]\nLet $b$ be a non-negative and $b'$ be any integer. By the previous claim, we have an isomorphism $\\d'\\colon\\K^{b,b'}\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}}\\L^{b,b'+1}$.\n\nBy symmetry, if $b'$ is non-negative and $b$ is any integer, we get a second isomorphism $\\d\\colon\\K^{b,b'}\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}}\\L^{b+1,b'}$. Thus, for any non-negative integer $k$, we obtain a zigzag of isomorphisms:\n\\[\n\\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}, scale=.6, shorten <=-2pt, shorten >=-2pt]\n\\scriptsize\n\\foreach \\i in {0,...,3} {\n  \\foreach \\j in {1,...,4} {\n    \\node (\\i\\j) at (\\i,\\j) {$\\bullet$};\n  }\n}\n\\node[fill=white] (01) at (0,1) {$\\K^{0,k}$};\n\\node[fill=white] (34) at (3,4) {$\\K^{k,0}$};\n\\begin{scope}[shift={(1,1)}]\n\\foreach \\i\/\\j\/\\k in {0\/1\/2,1\/2\/3,2\/3\/4} {\n  \\draw[-latex] (\\i\\j)--(\\j\\j);\n  \\draw[latex-] (\\j\\j)--(\\j\\k);\n}\n\\end{scope}\n\\end{tikzpicture}\n\\begin{tikzcd}\n\\K^{k,0} \\rar[\"\\sim\", \"\\d'\"']& \\L^{k,1} &\\lar[\"\\sim\"', \"-\\d\"] \\K^{k-1,1} \\rar[\"\\sim\", \"\\d'\"']& \\L^{k-1,2} &\\lar[\"\\sim\"', \"-\\d\"] \\cdots \\rar[\"\\sim\", \"\\d'\"']& \\L^{1,k} &\\lar[\"\\sim\"', \"-\\d\"] \\K^{0,k}.\n\\end{tikzcd} \\]\nNotice that we take $-\\d$ and not $\\d$. We refer to Remark \\ref{rem:preserve_cohomology} for an explanation of this choice.\n\n\\medskip\n\nSince $\\d'$ is injective on $\\AA^{\\bul,0}$, and since $\\Im(\\d')\\cap\\AA^{k,0}=\\{0\\}$, we obtain\n\\[ \\K^{k,0}=\\frac{\\ker(\\d\\d')\\cap\\AA^{k,0}}{\\bigl(\\Im(\\d)+\\Im(\\d')\\bigr)\\cap\\AA^{k,0}}=\\frac{\\ker(\\d)\\cap\\AA^{k,0}}{\\Im(\\d)\\cap\\AA^{k,0}}=H^k(\\AA^{\\bul,0}). \\]\nBy a symmetric argument, $\\K^{0,k}=H^k(\\AA^{0,\\bul})$. This gives an isomorphism\n\\[ H^k(\\AA^{\\bul,0})\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} H^k(\\AA^{0,\\bul}). \\]\n\nThe second part of the lemma is clear from the above claim and the preceding arguments. (In the case $\\Phi$ anticommutes, the commutative diagram holds for $(-1)^{b+b'}\\Phi$ which commutes.)\n\\end{proof}\n\n\\begin{remark} \\label{rem:preserve_cohomology}\nWe now explain why we choose the isomorphisms $\\d'$ and $-\\d$ in the above proof. Let $a\\in\\L^{k-i,i+1}$. The zigzag map sends $a$ to $b\\in\\K^{k-i,i}$ such that $\\d'b=a$, and also to $a'=-\\d b\\in\\L^{k-i+1,i}$. Thus, $a-a'=(\\d+\\d')b\\in\\Im(\\d+\\d')$. Moreover, by the definition of $\\L^{\\bul,\\bul}$, it is clear that $a$ and $a'$ belong to $\\ker(\\d+\\d')$. Hence, $a$ and $a'$ are two representatives of the same element of\n\\[ H^{k+1}\\bigl(\\Tot(\\AA^{\\bul,\\bul}, \\d+\\d')\\bigr), \\]\ni.e., the zigzag isomorphism is just identity on the cohomology of $\\AA^{\\bul,\\bul}$.\n\\end{remark}\n}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:spectral_isomorphism} in page one]\nDefine the double complex $\\AAa{a}^{b,b'}$ by shifting the double complex $\\Da{a}^{b,b'}$ by the vector $(1,1)$ and by inserting the the complexes $\\CCp{p}_0^{a,\\bullet}[1]$ and $\\STinf{p}_0^{a,\\bullet}[1]$ as the zero-th row and the zero-th column, respectively. More precisely, we set\n\\[ \\AAa{a}^{b,b'}=\\begin{cases}\n  0            & \\text{if $b<0$ or $b'<0$,} \\\\\n  \\CCp{p}_0^{a,b-1} & \\text{if $b'=0$} \\\\\n  \\STinf{p}_0^{a,b'-1} & \\text{if $b=0$} \\\\\n  \\Da{a}^{b-1,b'-1} & \\text{if $b>0$ and $b'>0$,}\n\\end{cases} \\]\nwith the corresponding differentials $\\d$ and $\\d'$ of respective bidegree $(1,0)$ and $(0,1)$ such that\n\\begin{align*}\n\\AAa{a}^{\\bul,0}  &= \\CCp{p}_0^{a,\\bul}[1], \\\\\n\\AAa{a}^{\\bul,b'} &= \\STinf{p}_0^{a,b'-1} \\hookrightarrow \\Da{a}^{\\bul,b'-1}[1] \\qquad \\forall b'\\geq 1, \\\\\n\\AAa{a}^{0,\\bul}  &= \\STinf{p}_0^{a,\\bul}[1],\\text{ and } \\\\\n\\AAa{a}^{b,\\bul}  &= \\CCp{p}_0^{a,b-1}\\hookrightarrow\\Da{a}^{b-1,\\bul}[1] \\qquad \\forall b\\geq 1.\n\\end{align*}\nOne can easily extend Proposition \\ref{prop:D_commuting} to get that $\\d$ and $\\d'$ commute (the details are given in Appendix~\\ref{sec:technicalities}). Moreover, by the exactness of Deligne sequence and by Proposition \\ref{prop:D_to_STinf}, all rows but the $0$-th one and all columns but the $0$-th one are exact. Thus we can apply Lemma \\ref{lem:zigzag} to get an isomorphism\n\\[ H^\\bul(\\AAa{a}^{\\bul,0})\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} H^\\bul(\\AAa{a}^{0,\\bul}). \\]\nLooking at the definition of $\\AAa{a}^{\\bul,\\bul}$, up to a shift by 1, this isomorphism is just what we want, i.e., the following isomorphism in page one.\n\\[ \\CCp{p}_1^{a,\\bul}\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} \\STinf{p}_1^{a,\\bul}. \\]\n\n\\medskip\n\nIt remains to prove the commutativity of the following diagram.\n\\[ \\begin{tikzcd}\n\\CCp{p}_1^{a,\\bul} \\dar[\"\\sim\"{above,sloped}]\\rar[\"\\i^*+\\pi^*\"]& \\CCp{p}_1^{a+1,\\bul} \\dar[\"\\sim\"{above,sloped}]\\\\\n\\STinfI{p}_1^{a,\\bul} \\rar[\"\\i^*+\\pi^*\"]& \\STinfI{p}_1^{a+1,\\bul}\n\\end{tikzcd} \\]\nTo prove this, we construct two anticommutative morphisms of double complexes.\n\nThe first morphism corresponds to $\\pi^*$ and is naturally defined as follows. On $\\CCp{p}^{a,b}$ it is given by $\\pi^*\\otimes\\id$, on $\\STinf{p}^{a,b'}$ it equals $\\pi^*$, and on $\\Da{a}^{b,b'}$ it is defined by $\\pi^*\\otimes\\id$. The anticommutativity properties are easy to check.\n\nThe second morphism, which we denote $\\d^\\i$, corresponds to $\\i^*$ and is defined as follows.\nTo keep the reading flow, we postpone the proof that $\\d^\\i$ is indeed a morphism and more details about the construction to Appendix~\\ref{sec:technicalities}.\n\n\\medskip\n\nFor each face $\\delta\\in X$, let $\\p_\\delta\\colon N^{\\sed(\\delta)}_\\mathbb{R} \\to \\TT_\\delta$ be a projection as defined just after Proposition \\ref{prop:grading1}. Assume moreover that the projections are compatible in the following sense:\n\\begin{itemize}\n\\item if $\\gamma\\ssubface\\delta$ have not the same sedentarity, then $\\pi_{\\sed(\\delta)\\ssubface\\sed(\\gamma)} \\p_\\delta=\\p_\\gamma \\pi_{\\sed(\\delta)\\ssubface\\sed(\\gamma)}$, where $\\pi_{\\sed(\\delta)\\ssubface\\sed(\\gamma)}: N^{\\sed(\\delta)}_{\\mathbb{R}}\\to N^{\\sed(\\gamma)}_{\\mathbb{R}}$ is the natural projection;\n\\item if $\\gamma\\prec\\delta$, then $\\p_\\gamma\\p_\\delta=\\p_\\gamma$.\n\\end{itemize}\n\nFor instance, we can choose an inner product on $N_\\mathbb{R}$ and extend it naturally on all the strata $N^\\sigma_\\mathbb{R}$ for $\\sigma\\in X_\\infty$. Then, if the projections $\\p_\\delta$ are chosen to be the orthogonal projections with respect to this inner product, then they are compatible in the above sense.\n\n\\medskip\n\nWith these conventions, we define $\\d^\\i$ on $\\STinf{p}^{a,b'}$ by $\\d^\\i:=\\i^*$. On $\\CCp{p}^{a,b}$, let $\\alpha\\otimes\\beta \\in \\bigwedge^{b}\\TT^\\dual\\gamma\\otimes\\SF^{p-b}(\\conezero^\\gamma)$, where $\\delta\\ssupface\\gamma$ is a pair of faces of same sedentarity and $\\dims\\gamma=a+b$. Then the part of its image by $\\d^\\i$ in $\\bigwedge^{b}\\TT^\\dual\\delta\\otimes\\SF^{p-b}(\\conezero^\\delta)$ is defined by\n\\begin{equation} \\label{eqn:d''}\n\\sign(\\gamma,\\delta) \\p_\\gamma\\rest{\\TT\\delta}^*(\\alpha) \\otimes (\\u \\mapsto \\pi_\\delta^*\\beta(\\u')),\n\\end{equation}\nwhere, for $\\u\\in\\SF^{p-b}(\\conezero^\\delta)$, the multivector $\\u'$ denotes the only element of $\\bigwedge^{p-b}\\ker(\\p_\\gamma)$ such that $\\pi_\\delta(\\u')=\\u$ (one can check that $\\u\\in \\SF_{p-b}(\\gamma)$).\n\n\\medskip\n\nOn $\\Da{a}^{b,b'}$, let $\\alpha\\otimes x \\in \\bigwedge^b\\TT^\\dual\\gamma \\otimes H^{2b'}(\\eta)$ where $\\eta\\succ\\gamma$ is a face of dimension $\\dims\\gamma+p-b-b'$. Then the part of its image by $\\d^\\i$ in $\\bigwedge^b\\TT^\\dual\\delta \\otimes H^{2b'}(\\mu)$, with $\\mu\\ssupface\\delta$, is given by\n\\[ \\sign(\\eta,\\mu) \\nvect^\\dual_{\\delta\/\\gamma}(\\p_\\delta(u))\\p_\\gamma\\rest\\delta^*(\\alpha) \\otimes \\i^*_{\\eta\\ssubface\\mu}(x), \\]\nwhere $u$ is a vector going from any point of $\\TT\\gamma$ to the vertex of $\\mu$ which is not in $\\eta$.\n\nWe will prove in Appendix~\\ref{sec:technicalities} that this map is indeed a morphism, and that the induced map on the first page $\\CCp{p}_1^{a,\\bul}$ is equal to the differential corresponding to $\\i^*$ given by the spectral sequence. This concludes the proof of the proposition.\n\\end{proof}\n\n\n\n\\subsection{Proof of Proposition~\\ref{prop:D_to_STinf}} \\label{sec:D_to_STinf} In order to conclude the proof of isomorphism between the first pages,\nwe are thus left to prove Proposition \\ref{prop:D_to_STinf}. This proposition claims the exactness of the following sequence for any integer $b'$:\n\\begin{align*}\n0 \\to&\n  \\STinfab\\eta{p+a-b'}{a}{2b'} \\to\n  \\Dab{a}{0}{p+a-b'}{2b'} \\to\n  \\Dab{a+1}{1}{p+a-b'}{2b'} \\to \\\\[1em] & \\hspace{3cm}\n  \\dots \\to\n  \\Dab{a+p-b'}{p-b'}{p+a-b'}{2b'} \\to\n0,\n\\end{align*}\nwhere we recall that the maps are $\\i^*_\\parr\\otimes\\id$ and that the relations $\\eta \\succ \\delta$ only concern those faces which have the same sedentarity $\\sed(\\eta)=\\sed(\\delta)$.\n\n\\medskip\n\nAs the form of each term suggests, we can decompose this sequence as a direct sum of sequences over fixed $\\eta$ of dimension $s:=p+a-b'$ as follows:\n\\[ \\bigoplus_{\\dims\\eta=s}\\ \\Bigl(\n  0 \\to \\mathbb{R}^{\\epsilon_\\eta} \\to \\Sxab{a}0 \\to \\Sxab{a+1}1 \\to \\dots \\to \\Sxab{s}{p-b'} \\to 0\n\\Bigr)\\otimes H^{2b'}(\\eta), \\]\nwhere\n\\[ \\epsilon_\\eta=\\begin{cases}\n  1 & \\text{if $\\dims{\\eta_\\infty}\\leq a$,} \\\\\n  0 & \\text{otherwise.}\n\\end{cases} \\]\n\nTherefore, the following result clearly implies Proposition~\\ref{prop:D_to_STinf}.\n\n\\begin{prop}\nFor each pair of non-negative integers $r \\leq n$, and for each (simplicial) unimodular polyhedron $\\eta$ of dimension $n$, the cochain complex\n\\[ \\Cx\\eta{r}\\colon 0 \\longrightarrow \\Sxab{r}0[r] \\longrightarrow \\Sxab{r+1}1[r+1] \\longrightarrow \\dots \\longrightarrow \\Sxab{n}{n-r}[n] \\longrightarrow 0, \\]\nwhose maps are given by $\\i^*_\\parr$, has cohomology\n\\[ \\begin{cases}\n  \\mathbb{R}[r] & \\text{if $\\dims{\\eta_\\infty}\\leq r$}, \\\\\n  0 & \\text{if $\\dims{\\eta_\\infty}>r$.}\n\\end{cases} \\]\n\\end{prop}\n\\begin{proof} We will reduce the statement to the computation of some appropriate cohomology groups of the tropical projective spaces.\n\nWe begin with the case $\\dims{\\eta_\\infty}=0$, i.e., $\\eta$ is a unimodular simplex. Let us study the dual complex instead, which using the duality between $\\bigwedge^r\\TT^\\dual\\delta $ and $\\bigwedge^{\\dims{\\delta}-r} \\TT^\\dual\\delta$ for each simplex $\\delta$, has the following form\n\\[ 0 \\longrightarrow \\Sxab{n}r[n] \\longrightarrow \\Sxab{n-1}r[n-1] \\longrightarrow \\dots \\longrightarrow \\Sxab{r}r[r] \\longrightarrow 0. \\]\nThe maps in this complex are induced by restriction map $\\bigwedge^r\\TT^\\dual\\delta \\to \\bigwedge^r\\TT^\\dual\\zeta$ for inclusion of faces $\\zeta \\ssubface \\delta$, extended with $\\sign$ according to our convention in Section \\ref{sec:basic_maps}.\n\n\\medskip\n\nFor the ease of arguments, it will be convenient to choose an inner product $\\pairing{\\,\\cdot\\,}{\\cdot\\,}$ on $\\TT\\eta$. This inner product restricts to $\\TT\\delta$ for each face $\\delta$ and extends naturally to multivectors. More precisely, for any collection of vectors $u_1, \\dots, u_k, v_1, \\dots, v_k$, we set\n\\[ \\pairing{u_1\\wedge\\dots\\wedge u_k}{v_1\\wedge\\dots\\wedge v_k} := \\det\\Bigl(\\,\\bigl(\\pairing{u_i}{v_j}\\bigr)_{1\\leq i,j\\leq k}\\,\\Bigr). \\]\n\nLet $\\gamma \\ssubface\\delta$ be a pair of faces. We define $\\p_{\\delta\\ssupface\\gamma}\\colon \\TT\\delta\\to\\TT\\gamma$ to be the orthogonal projection and naturally extend it to multivectors in the exterior algebra.\n\n\\medskip\n\nLet $\\alpha\\in\\bigwedge^r\\TT^\\dual\\delta$. We denote by $\\alpha^\\dual$ the dual multivector of $\\alpha$, defined by the property\n\\[ \\alpha( \\u ) = \\pairing{\\alpha^\\dual}{\\u} \\]\nfor any $\\u \\in \\bigwedge^r\\TT\\delta$.\n\nMoreover, we denote by $\\alpha\\rest\\gamma$ the restriction of $\\alpha$ on $\\bigwedge^r\\TT\\gamma$. By adjunction property, we get the following equality:\n\\[ \\p_{\\delta\\ssupface\\gamma}(\\alpha^\\dual)=(\\alpha\\rest\\gamma)^\\dual. \\]\nIndeed, for any $\\u\\in\\bigwedge^r\\TT\\gamma\\subset\\bigwedge^r\\TT\\delta$, we have\n\\[ \\alpha\\rest\\gamma(\\u) = \\alpha(\\u) = \\pairing{\\alpha^\\dual}\\u = \\pairing{\\p_{\\delta\\ssupface\\gamma}(\\alpha^\\dual)}\\u. \\]\n\n\\medskip\n\nThis implies that the linear map which sends $\\alpha\\mapsto\\alpha^\\dual$ provides an isomorphism between the two following complexes: our original complex on one side,\n\\[ \\begin{tikzcd}[column sep=small]\n0 \\rar& \\Sxab{n}r \\rar& \\Sxab{n-1}r \\rar& \\raisebox{2pt}[8pt][4pt]{\\makebox[20pt]{$\\dots$}} \\rar& \\Sxab{r}r \\rar& 0,\n\\end{tikzcd} \\]\nand its dual\n\\[ \\begin{tikzcd}[column sep=small]\n0 \\rar& \\Syab{n}r \\rar& \\Syab{n-1}r \\rar& \\raisebox{2pt}[8pt][4pt]{\\makebox[20pt]{$\\dots$}} \\rar& \\Syab{r}r \\rar& 0.\n\\end{tikzcd} \\]\nNote that the maps of the second complex are given by the orthogonal projections $\\p$.\n\n\\medskip\n\nWe claim that the second complex above computes $H^{r,\\bul}_\\trop(\\TP^n)$, which leads the result. Indeed, the only non-trivial cohomology groups of $\\TP^n$ are $H^{p,p}_\\trop (\\TP^n)\\simeq \\mathbb{R}$ for $0\\leq p\\leq n$.\n\n\\medskip\n\nDenote by $\\delta_0, \\dots, \\delta_n$ all the faces of codimension one of $\\eta$. For each $\\delta_i$, let $\\ell_i$ be the affine map on $\\eta$ which is identically equal to $1$ on $\\delta_i$ and takes value $0$ on the opposite vertex to $\\delta_i$. Let $\\ell_i^0\\in\\TT^\\dual\\eta$ be the linear map corresponding to $\\ell_i$. It is easy to see that $\\sum_i\\ell_i^0=0$. Set $u_i:=(\\ell_i^0)^\\dual\\in\\TT\\eta$. Let $(e_0, \\dots, e_n)$ be the standard basis of $\\mathbb{R}^n$. We get a linear isomorphism $\\TT\\eta\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} N_\\mathbb{R}=\\rquot{\\mathbb{R}^{n+1}}{(1,\\dots,1)}$ by mapping the $u_i$ to $e_i$ seen in $N_\\mathbb{R}$. The family $(e_i)_i$ induce a natural compactification of $N_\\mathbb{R}$ into $\\TP^n$, and each face $\\delta$ of $\\eta$ corresponds to a stratum $N_{\\mathbb{R}, \\delta}$ of dimension $\\dims\\delta$. Moreover, the linear isomorphism $\\TT\\eta\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} N_\\mathbb{R}$ induces a linear isomorphism $\\TT\\delta\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} N_{\\mathbb{R},\\delta}$. The projections $\\pi$ on $\\TP^n$ and $\\p$ on $\\eta$ commute with these isomorphisms. Thus we get an isomorphism between complexes:\n\\[ \\begin{tikzcd}[column sep=small]\n0 \\rar& \\Syab{n}r \\rar& \\Syab{n-1}r \\rar& \\raisebox{2pt}[8pt][4pt]{\\makebox[20pt]{$\\dots$}} \\rar& \\Syab{r}r \\rar& 0, \\\\\n0 \\rar& \\SF^r(N_\\mathbb{R}) \\rar& \\displaystyle\\bigoplus_{\\delta\\prec\\eta \\\\ \\dims\\delta = n-1}\\SF^r(N_{\\mathbb{R},\\delta}) \\rar& \\raisebox{2pt}[8pt][4pt]{\\makebox[20pt]{$\\dots$}} \\rar& \\displaystyle\\bigoplus_{\\delta\\prec\\eta \\\\ \\dims\\delta = r}\\SF^r(N_{\\mathbb{R},\\delta}) \\rar& 0.\n\\end{tikzcd} \\]\nThe second complex is just the tropical simplicial complex of $\\TP^n$ for the coarsest simplicial subdivision. Hence, its cohomology is $H^{r,\\bul}_\\trop(\\TP^n)=\\mathbb{R}[r]$. These concludes the case $\\dims{\\eta_\\infty}=0$.\n\n\\medskip\n\nIt remains to generalize to any simplicial polyhedron $\\eta$. If $\\eta=\\rho$ is a ray, by a direct computation, the proposition holds: $H^\\bul(\\Cx\\rho{0})$ is trivial and $H^\\bul(\\Cx\\rho{1})=\\mathbb{R}[1]$. In general, $\\eta$ is isomorphic to $\\eta_\\f\\times\\underbrace{\\rho\\times \\dots \\times \\rho}_{m \\textrm{ times }}$, where $\\rho$ is a ray and $m:=\\dims{\\eta_\\infty}$. Moreover,\n\\[ \\Cx\\eta{r}\\simeq\\bigoplus_{r_0,\\dots,r_m\\geq 0 \\\\ r_0+\\dots+r_m=r} \\Cx{\\eta_\\f}{r_0}\\otimes\\Cx\\rho{r_1}\\otimes\\dots\\otimes\\Cx\\rho{r_m}. \\]\nBy the K\u00fcnneth formula for cochain complexes of vector spaces, this decomposition also holds for cohomology. I.e., we have\n\\[ H^\\bul(\\Cx\\eta{r})\\simeq\\bigoplus_{r_0,\\dots,r_m\\geq0 \\\\ r_0+\\dots+r_m=r} H^\\bul(\\Cx{\\eta_\\f}{r_0})\\otimes H^\\bul(\\Cx\\rho{r_1})\\otimes\\dots\\otimes H^\\bul(\\Cx\\rho{r_m}). \\]\nLooking at the cohomology of the rays, all terms are trivial except the term of indices $r_1=\\dots=r_m=1$ and $r_0=r-m$ if $m\\leq r$. In this case, we get\n\\begin{align*}\nH^\\bul(\\Cx\\eta{r})\n  &\\simeq H^\\bul(\\Cx{\\eta_\\f}{r-m})\\otimes H^\\bul(\\Cx\\rho1)^{\\otimes m} \\\\\n  &\\simeq \\mathbb{R}[r-m]\\otimes\\mathbb{R}[1]^{\\otimes m} \\\\\nH^\\bul(\\Cx\\eta{r}) &= \\mathbb{R}[r].\n\\end{align*}\nThis concludes the proof of the proposition.\n\\end{proof}\n\nWe have so far proved that the first pages of $\\CCp{p}^{\\bul,\\bul}$ and of $\\STinf{p}^{\\bul,\\bul}$ coincide. Unfortunately, this is not sufficient in general to conclude that the two spectral sequences have the same abutment. Thus, we have to extend the isomorphism to further pages. A direct use of the zigzag lemma on further pages is tedious. Therefore, in the next section, we present a proof using a kind of generalization of the zigzag lemma.\n\n\n\n\\subsection{Isomorphisms on further pages}\n\\label{sec:every_pages}\n\n{\\renewcommand{\\DI}{\\prescript{\\textup I}{}\\textnormal{\\textsf{D}}}\n\\newcommand{\\prescript{\\textup I}{}\\partial}{\\prescript{\\textup I}{}\\partial}\n\\newcommand{\\prescript{\\textup I}{}C}{\\prescript{\\textup I}{}C}\n\\newcommand{\\CCpI}[1]{\\prescript{\\textup I}{#1}\\textnormal{\\textsf{C}}}\n\nIn this section, we achieve the proof of Theorem \\ref{thm:spectral_isomorphism}. To do so, start by stating a general lemma about spectral sequences which we call the \\emph{spectral resolution lemma}. Roughly speaking, this lemma states that, under some natural assumptions, a \\emph{resolution} of a spectral sequence results in another spectral sequence which coincides with the first one on any further pages.\n\nTo prove Theorem \\ref{thm:spectral_isomorphism}, we will apply this lemma to the \\emph{triply indexed differential complex} $(\\Da{\\bul}^{\\bul,\\bul}, \\partial=\\d^\\i+\\pi^*\\d+\\d')$ introduced above (actually a slight modification obtained by inserting the Steenbrink and tropical sequences). This triply indexed complex plays the role of a bridge between the two spectral sequences $\\CCp{p}^{\\bul,\\bul}$ and $\\STinfI{p}^{\\bul,\\bul}$. In fact, as we will show below, it provides a \\emph{spectral resolution}, in a sense which will be made precise in a moment, of both the spectral sequences at the same time. Thus, applying the spectral resolution lemma allows to directly conclude the proof of Theorem \\ref{thm:spectral_isomorphism}.\n\n\\medskip\n\nThough the method presented below gives an alternate proof of the isomorphism in page one, independent of the one given in the previous section, the intermediate results of the previous section will be crucial in showing the stated resolution property, and so to verify the assumption of the spectral resolution lemma. In a sense, this lemma stands upon the basic case treated in the previous section. Moreover, as the proof of the spectral resolution lemma shows, this lemma can be regarded as a generalization of the zig-zag lemma used in the previous section.\n\n\n\\subsubsection{Triply indexed differential complexes} Before we state the lemma, we introduce some conventions. By a \\emph{triply indexed differential complex $(\\Enop^{\\bul,\\bul,\\bul}, \\partial)$}, we mean\n\\begin{itemize}\n\\item $\\Enop^{a,b,c}=0$ unless $a, b, c\\geq0$;\n\\item the map $\\partial\\colon \\Enop^{\\bul,\\bul,\\bul} \\to \\Enop^{\\bul,\\bul,\\bul}$ is of degree one, i.e., it can be decomposed as follows:\n\\begin{gather*}\n\\partial=\\bigoplus_{i,j,k\\in\\mathbb{Z} \\\\ i+j+k=1}\\partial^{i,j,k}, \\text{ with} \\\\\n\\forall i,j,k\\in\\mathbb{Z}, \\qquad \\partial^{i,j,k} = \\bigoplus_{a,b,c\\in\\mathbb{Z}}\\partial^{i,j,k}\\rest{\\Enop^{a,b,c}},\n\\end{gather*}\nwhere $\\partial^{i,j,k}\\rest{\\Enop^{a,b,c}}$ is a map from $\\Enop^{a,b,c}$ to $\\Enop^{a+i,b+j,c+k}$;\n\\item we have $\\partial\\partial=0$.\n\\end{itemize}\n\n\\medskip\n\nWith these assumptions, the total complex $\\Tot^\\bul(E^{\\bul,\\bul,\\bul})$ becomes a differential complex.\n\n\\medskip\n\nThe \\emph{filtration induced by the first index} on $E^{\\bul, \\bul,\\bul}$ is by definition the decreasing filtration\n\\[ \\Enop^{\\bul,\\bul,\\bul} = \\Enop^{\\geq 0,\\bul,\\bul} \\supseteq \\Enop^{\\geq 1,\\bul,\\bul} \\supseteq \\Enop^{\\geq 2,\\bul,\\bul} \\supseteq \\dots, \\]\nwhere\n\\[ \\Enop^{\\geq a,\\bul,\\bul} = \\bigoplus_{a'\\geq a}\\Enop^{a',\\bul,\\bul}. \\]\nNotice that the differential $\\partial$ preserves the filtration induced by the first index if and only if $\\partial^{i,j,k}=0$ for any $i<0$. In such a case, we denote by $\\prescript{\\textup I}{}\\Enop^{\\bul,\\bul}_0$ the 0-th page of the induced spectral sequence abutting to the cohomology of the total complex $\\Tot^\\bul(E^{\\bul,\\bul,\\bul})$. It has the following shape:\n\\[ \\prescript{\\textup I}{}\\Enop^{a,b}_0 = \\Tot^{a+b}(\\Enop^{a,\\bul,\\bul})  = \\bigoplus_{m, n\\\\ m+n=b} \\Enop^{a, m,n}\\]\nendowed with the differential\n\\[ \\prescript{\\textup I}{}\\partial := \\sum_{k,l\\in\\mathbb{Z}}\\partial^{0,j,k}. \\]\nThe $k$-th page of this spectral is denoted by $\\prescript{\\textup I}{}\\Enop^{\\bul,\\bul}_k$.\n\n\\medskip\n\nWe use analogous conventions as above to define bi-indexed differential complexes.\n\n\n\\subsubsection{Statement of the spectral resolution lemma} Let $(C^{\\bul,\\bul}, \\d)$ be a bi-indexed differential complex. We assume that $\\d$ preserves the filtration induced by the first index, i.e., we assume $\\d^{i,j}=0$ if $i<0$.\n\n\\medskip\n\nA \\emph{spectral resolution} of $(C^{\\bul,\\bul}, \\d)$ is a triply indexed differential complex $(\\Enop^{\\bul,\\bul,\\bul}, \\partial)$ which verifies the following properties enumerated R1-R4:\n\n\\smallskip\n\\begin{enumerate}[label=(R\\arabic*)]\n\\item \\label{enum:R1} There is an inclusion $\\i\\colon C^{\\bul,\\bul}\\to\\Enop^{\\bul,\\bul,0} \\subseteq \\Enop^{\\bul,\\bul,\\bul}$ which respects the bi-indices, i.e., such that $\\i\\d=\\partial\\i$ (here $\\partial$ is \\emph{not} restricted to $\\Enop^{\\bul,\\bul,0}$).\n\n\\smallskip\n\\item \\label{enum:R2} The differential $\\partial$ of $E^{\\bul,\\bul,\\bul}$ preserves the filtration induced by the first index, i.e., we have $\\partial^{i,j,k}=0$ if $i<0$.\n\n\\smallskip\n\\item \\label{enum:R3} We have $\\partial^{i,j,k}=0$ if $k\\geq 2$.\n\\end{enumerate}\n\n\\medskip\n\nDenote by $\\partial_3:=\\partial^{0,0,1}$. From the fact that $\\partial$ is a differential and the assumptions above, we deduce that $\\partial_3\\partial_3=0$. Finally, we assume the following resolution property.\n\n\\smallskip\n\\begin{enumerate}[resume*]\n\\item \\label{enum:R4} For any $a,b$ in $\\mathbb{Z}$, the differential complex $(\\Enop^{a,b,\\bul}, \\partial_3\\rest{\\Enop^{a,b,\\bul}})$ is a right resolution of $C^{a,b}$.\n\\end{enumerate}\n\n\\medskip\n\nDenote by $\\prescript{\\textup I}{}\\Enop^{\\bul,\\bul}$ and $\\prescript{\\textup I}{}C^{\\bul,\\bul}$ the spectral sequences induced by the filtration by the first index on $E^{\\bul,\\bul,\\bul})$. Then the inclusion $\\i$ induces canonical compatible isomorphisms\n\\[ \\prescript{\\textup I}{}C_k^{\\bul,\\bul} \\simeq \\prescript{\\textup I}{}\\Enop_k^{\\bul,\\bul} \\]\nbetween $k$-th pages for any $k\\geq1$.\n\n\\begin{lemma}[Spectral resolution lemma] \\label{lem:spectral_resolution}\nLet $(C^{\\bul,\\bul}, \\d)$ be a bi-indexed complex such that the differential $\\d$ preserves the filtration induced by the first index. Let $(\\Enop^{\\bul,\\bul,\\bul}, \\partial)$ be a spectral resolution of $C^{\\bul, \\bul}$.\n\nDenote by $\\prescript{\\textup I}{}\\Enop^{\\bul,\\bul}$ and $\\prescript{\\textup I}{}C^{\\bul,\\bul}$ the spectral sequences induced by the filtration by the first indices on $E^{\\bul,\\bul,\\bul}$ and $C^{\\bul, \\bul}$, respectively. Then, the inclusion $\\i \\colon C^{\\bul,\\bul} \\hookrightarrow E^{\\bul, \\bul, \\bul}$ induces canonical compatible isomorphisms\n\\[ \\prescript{\\textup I}{}C_k^{\\bul,\\bul} \\simeq \\prescript{\\textup I}{}\\Enop_k^{\\bul,\\bul} \\]\nbetween $k$-th pages of the two spectral sequences for any values of $k\\geq1$.\n\\end{lemma}\n\nWe assume for the moment this lemma and finish the proof of Theorem \\ref{thm:spectral_isomorphism}.\n\n\n\\subsubsection{Proof of Theorem \\ref{thm:spectral_isomorphism}}\nWe use the notations of Section~\\ref{sec:CC_to_STinf}. We will apply the spectral resolution lemma twice in order to achieve the isomorphism of pages staged in the theorem.\n\n\\medskip\n\nWe start by gathering the double complexes $\\Da{a}^{\\bul,\\bul}$ together for all $a$ in order to construct a triply indexed differential complex $(\\Dnop^{\\bul,\\bul,\\bul}, \\partial)$ as follows. First, set\n\\[ \\Dnop^{a,b,b'} := \\Da{a}^{b,b'} \\]\nand let the two differentials of multidegrees $(0,1,0)$ and $(0,0,1)$ be equal to the differentials $\\d$ and $\\d'$ of the double complexes $\\Da{a}^{\\bul,\\bul}$.\n\nIn the course of proving the isomorphism between the first pages of the spectral sequences in Theorem~\\ref{thm:spectral_isomorphism}, we introduced in Section~\\ref{sec:CC_to_STinf} two anticommutative morphisms $\\pi^*$ and $\\d^\\i$ of multidegree $(1,0,0)$ from $\\Da{a}^{\\bul,\\bul}\\to \\Da{a+1}^{\\bul,\\bul}$.\n\n\\medskip\n\nWe set $\\partial =\\d^\\i+ \\pi^* + \\d+ \\d'$.\n\n\\begin{prop}\\label{prop:differential_triple_D}\nWe have $\\partial \\circ \\partial =0$.\n\\end{prop}\n\\begin{proof}\nThe proof consists of calculating directly all the terms which appear in this decomposition. This will be explained in Appendix~\\ref{sec:technicalities}.\n\\end{proof}\n\nDenote by $\\CCp{p}^{\\bul,\\bul}$ the bi-indexed complex $\\CCp{p}_0^{\\bul,\\bul}$ which is endowed with the differential $\\d+\\d'$, where $\\d$ and $\\d'$ are the maps defined on $\\AA^{\\bul,0}=\\Tot^\\bul(\\CCp{p}^{\\bul,\\bul})$ in the previous section that we restrict to its zero-th row.\n\nThe inclusion $\\AA^{\\bul,0}\\to\\AA^{\\bul,1}$ gives an inclusion of complexes of $\\CCp{p}^{\\bul,\\bul}$ into $\\Da{p}^{\\bul,\\bul,\\sqbullet}$. Moreover, for any integers $a$ and $b$, the complex $(\\Dnop^{a,b,\\sqbullet}, \\d')$ is the Deligne resolution of $\\CCp{p}_0^{a,b}$. Thus we can apply the spectral resolution lemma. This gives canonical isomorphisms\n\n\\begin{equation} \\label{eq:iso_pages_1}\n\\forall \\,\\,\\, k\\geq 1, \\quad \\CCpI{p}_k^{\\bul,\\bul} \\simeq \\DI_k^{\\bul,\\bul}.\n\\end{equation}\n\nA priori, $\\CCpI{p}_k^{\\bul,\\bul}$ could be different from $\\CCp{p}_k^{\\bul,\\bul}$. So it might appear somehow surprising to see that they are actually equal thanks to the following proposition.\n\\begin{prop} \\label{prop:isomorphism_filtrations}\nThe natural isomorphism $C^{p,\\bul}_\\trop \\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} \\Tot^\\bul(\\CCp{p}^{\\bul,\\bul})$ is an isomorphism of filtered differential complexes. Here $C^{p,\\bul}_\\trop$ comes with differential $\\partial_\\trop$ and weight filtration $W^\\bul$, the differential on $\\Tot^\\bul(\\CCp{p}^{\\bul,\\bul})$ is $\\d+\\d'$ and the filtration is induced by the first index.\n\\end{prop}\n\\begin{proof}\nThe proof of this proposition is given in Appendix~\\ref{sec:technicalities}.\n\\end{proof}\n\nIn the same way, from Section~\\ref{sec:CC_to_STinf}, we deduce an inclusion of complexes of $\\STinf{a}^{\\bul,\\bul}$, with the usual differentials $\\gys$ and $\\i^*+\\pi^*$, into $\\Dnop^{\\bul,0,\\bul}$. Moreover, by Proposition~\\ref{prop:D_to_STinf}, proved in Section~\\ref{sec:D_to_STinf}, for any pair of integers $a$ and $b'$, the complex $(\\Dnop^{a,\\blacktriangle,b'}, \\d)$ is a resolution of $\\STinf{p}_0^{a,b'}$.\n\n\\medskip\n\nThus, by applying again the spectral resolution lemma, we get canonical isomorphisms\n\n\\begin{equation} \\label{eq:iso_pages_2}\n\\forall \\,\\, k\\geq 1, \\quad \\STinfI{p}_k^{\\bul,\\bul} \\simeq \\DI_k^{\\bul,\\bul}.\n\\end{equation}\n\n\\medskip\n\nFrom \\eqref{eq:iso_pages_1} and \\eqref{eq:iso_pages_2}, we infer canonical isomorphisms\n\n\\[ \\CCp{p}_k^{\\bul,\\bul} \\simeq \\STinfI{p}_k^{\\bul,\\bul} \\]\nbetween $k$-th pages for any $k\\geq1$ of the two spectral sequences, which concludes the proof of Theorem \\ref{thm:spectral_isomorphism}. \\QED\n\n\n\\subsubsection{Proof of the spectral resolution lemma}  In this section we prove the resolution lemma. Assume that $\\i\\colon C^{\\bul, \\bul} \\hookrightarrow \\AA^{\\bul,\\bul, 0}\\subseteq \\AA^{\\bul,\\bul, \\bul}$ is a spectral resolution of the bi-indexed differential complex $C^{\\bul, \\bul}$.\n\n\\begin{proof}[Proof of Lemma \\ref{lem:spectral_resolution}]\nLet $k\\geq1$. By construction of the pages of the spectral sequences, we have\n\\[ \\prescript{\\textup I}{}\\Enop_k^{a,b}=\\frac{\\Bigl\\{x\\in\\Tot^{a+b}(\\Enop^{\\bul,\\bul,\\bul})\\cap\\Enop^{\\geq a,\\bul,\\bul} \\st \\partial x \\in \\Enop^{\\geq a+k,\\bul,\\bul}\\Bigr\\}}{\\Enop^{\\geq a+1,\\bul,\\bul} + \\partial\\Enop^{\\geq a-k+1,\\bul,\\bul}}. \\]\n(To be more rigorous, we should replace here the denominator by the intersection of the numerator with the denominator.)\n\n\\medskip\n\nThe idea is to prove that one can take a representative of the above quotient in the image of $\\i$. To do so, we use the following induction which can be seen as a generalization of the zigzag lemma.\n\n\\begin{claim} Let $a$ be an integer. The following quotient\n\\[ \\frac{\\Bigl\\{x\\in\\Tot^{a+b}(\\Enop^{\\bul,\\bul,\\bul})\\cap\\Enop^{\\geq a,\\bul,\\bul} \\st \\partial x \\in \\Enop^{\\geq a+k,\\bul,\\bul}\\Bigr\\}}  {\\Enop^{\\geq a+k,\\bul,\\bul} + \\partial\\Enop^{\\geq a-k+1,\\bul,\\bul} + \\i(C^{\\geq a,\\bul})}\\]\nis zero.\n\\end{claim}\n\nNotice that, in the denominator, we did not use $\\Enop^{\\geq a+1,\\bul,\\bul}$ but $\\Enop^{\\geq a+k,\\bul,\\bul}$. Doing so we get a stronger result which will be needed later.\n\n\\medskip\n\nLet $x$ be an element of the numerator of the following quotient. We first explain how to associate a vector in $\\mathbb{Z}^2$ to any $x$ in the numerator. Endowing $\\mathbb{Z}^2$ with the lexicographic order, and proceeding by induction on the lexicographical order of vectors associated to the elements $x$, we show that $x$ is zero in the quotient.\n\n\\medskip\n\nWe decompose $x$ as follows\n\\[ x = \\hspace{-2ex} \\sum_{\\alpha,\\beta,c\\in\\mathbb{Z} \\\\ \\alpha\\geq a \\\\ \\alpha+\\beta+c=a+b} \\hspace{-2ex} x_{\\alpha,\\beta, c}, \\qquad x_{\\alpha,\\beta,c}\\in\\Enop^{\\alpha,\\beta,c}. \\]\nLet $\\alpha$ be smallest integer such that there exists $\\beta,c\\in\\mathbb{Z}$ with $x_{\\alpha,\\beta,c}\\neq 0$. Note that if no such $\\alpha$ exists, then $x=0$ and we are done. Having chosen $\\alpha$, let now $c$ be the largest integer such that $x_{\\alpha,\\beta,c}\\neq0$ for $\\beta=a+b-\\alpha-c$. We associate to $x$ the vector $(-\\alpha, c)$.\n\n\\medskip\n\nIn order to prove the claim for a fixed value of $a$, we proceed by induction on the lexicographic order of the associated vectors $(-\\alpha, c)$. Given a vector $(-\\alpha, c)$, in what follows, we set $\\beta:=a+b-\\alpha-c$.\n\n\\smallskip\n\\emph{Base of the induction: $\\alpha\\geq a+k$.} Writing $x =\\sum x_{m,n,p}$ for $x_{m,n,p}\\in E^{m,n,p}$ we see by the choice of $\\alpha$ that $m \\geq \\alpha \\geq a+k$ for all non-zero terms $x_{m,n,p}$ it follows that all these terms belong to $\\Enop^{\\geq a+k,\\bul,\\bul}$, and so $x$ obviously belongs to the denominator.\n\n\\medskip\n\n\\emph{first case: $\\alpha<a+k$ and $c>0$.} Denote by $y=\\sum y_{\\alpha',\\beta',c'}$ the boundary $\\partial x$ of $x$. Then, since $\\partial x$ belongs to $\\Enop^{\\geq a+k,\\bul,\\bul}$, the element $y_{\\alpha,\\beta,c+1}$ must be zero. Moreover, since $\\partial^{a',b',c'}=0$ if $c'\\geq2$ or $a'<0$, the part $y_{\\alpha,\\beta,c+1}$ of $y$ must be equal to $\\partial_3( x_{\\alpha,\\beta,c})$. Thus $x_{\\alpha,\\beta,c}$ is in the kernel of $\\partial_3$. Since $(\\Enop^{\\alpha,\\beta,\\bul}, \\partial_3)$ is a resolution, we can take a preimage $z$ of $x_{\\alpha,\\beta,c}$ in $\\Enop^{\\alpha,\\beta,c-1}$ for $\\partial_3$. Then\n\\[ \\partial z \\in \\partial\\Enop^{\\geq a,\\bul,\\bul} \\subseteq \\partial\\Enop^{\\geq a-k+1,\\bul,\\bul}. \\]\nThus, $x$ is equivalent to $x'=x-\\partial z$, the vector associated to $x'$ is strictly smaller than $(-\\alpha, c)$, and we conclude again by the the hypothesis of our induction for $x'$.\n\n\\medskip\n\n\\emph{second case: $\\alpha<a+k$ and $c=0$}. We can follow the arguments used in the second case above up to the moment $x_{\\alpha,\\beta,0}\\in\\ker(\\partial_3)$. Then, $x_{\\alpha,\\beta,0}$ belongs to $\\i(C^{\\alpha,\\bul})$. Since $\\i(C^{\\geq a,\\bul})$ is in the denominator, we can subtract $x_{\\alpha,\\beta,0}$ from $x$ and conclude by induction.\n\n\\smallskip\nIn any case, we have proved $x$ is equal to zero and the claim follows.\nAs a consequence, we infer that\n\\begin{equation} \\label{eqn:EI2}\n\\prescript{\\textup I}{}\\Enop_k^{a,b} = \\frac{\\Bigl\\{x\\in\\Tot^{a+b}(\\Enop^{\\bul,\\bul,\\bul})\\cap\\Enop^{\\geq a,\\bul,\\bul} \\st \\partial x \\in \\Enop^{\\geq a+k,\\bul,\\bul}\\Bigr\\} \\cap \\Im(\\i)}  {\\bigl(\\Enop^{\\geq a+1,\\bul,\\bul} + \\partial\\Enop^{\\geq a-k+1,\\bul,\\bul}\\bigr) \\cap \\Im(\\i)}.\n\\end{equation}\n\n\\medskip\n\nThis has to be compared with\n\\[ \\prescript{\\textup I}{}C_k^{a,b} = \\frac{\\Bigl\\{z\\in\\Tot^{a+b}(C^{\\bul,\\bul})\\cap C^{\\geq a,\\bul} \\st \\d z \\in C^{\\geq a+k,\\bul}\\Bigr\\}}  {C^{\\geq a+1,\\bul} + \\d C^{\\geq a-k+1,\\bul}}. \\]\nFrom $\\i\\d=\\partial\\i$, we get that the induced map $\\i\\colon \\prescript{\\textup I}{}\\Enop_k^{a,b}\\to \\prescript{\\textup I}{}C_k^{a,b}$ is well-defined. Moreover, it is now clear that this map is surjective. For the injectivity, we have to work a bit more on the denominator of $\\prescript{\\textup I}{}\\Enop_k^{a,b}$.\n\n\\medskip\n\nLet $x$ be an element of the intersection of the denominator with the numerator of \\eqref{eqn:EI2}. In particular, we have $x\\in\\Enop^{\\geq a,\\bul,\\bul}$. Let $z\\in\\Enop^{\\geq a-k+1,\\bul,\\bul}$ be such that $x-\\partial z\\in\\Enop^{\\geq a+1,\\bul,\\bul}$.\n\n\\medskip\n\nNow we show that we can reduce to the case $z\\in\\Im(\\i)+\\Enop^{\\geq a+1,\\bul,\\bul}$. Notice that $\\partial z\\in E^{\\geq a,\\bul,\\bul}$. Thus, $z$ belongs to the numerator of $\\prescript{\\textup I}{}\\Enop_{k-1}^{a-k+1,b+k-2}$. If $k\\geq 2$, we can apply the induction above to get that $z=z_1+\\partial z_2+\\i z_3$ with $z_1\\in\\Enop^{\\geq a,\\bul,\\bul}$, $z_2\\in\\Enop^{\\geq a-2k+3,\\bul,\\bul}$ and $z_3\\in C^{\\geq a-k+1,\\bul}$. This decomposition also holds for $k=1$ setting $z_1=z$ and $z_2=z_3=0$. Since we are only concerned by $\\partial z$, we can assume without loss of generality that $z_2=0$.\n\nIf $z_1\\not\\in E^{\\geq a+1,\\bul,\\bul}$, let $c$ be the maximum integer such that $z_{a,\\beta,c}\\neq0$ for $\\beta=b-1-c$. Denote by $y=\\sum y_{\\alpha',\\beta',c'}$ the boundary $\\partial z$ of $z$. Since $\\partial\\i z_3\\in\\Enop^{\\bul,\\bul,0}$, we get that $y_{a,\\beta, c+1}=\\partial_3 \\, z_{a,\\beta,c}$. Moreover $x-y=x-\\partial z\\in\\Enop^{\\geq a+1,\\bul,\\bul}$. Hence, $x_{a,\\beta,c+1}$ must be equal to $y_{a,\\beta,c+1}$. Since $x\\in\\Im(\\i)\\subseteq\\Enop^{\\bul,\\bul,0}$, $x_{a,\\beta,c+1}=0$. From this we get $y_{a,\\beta,c+1}=\\partial_3z_{a,\\beta,c}=0$. Therefore, $z_{a,\\beta,c}$ is in the kernel of $\\partial_3$. Two cases can occur.\n\n\\begin{itemize}\n\\item If $c>0$, then we can find $w\\in \\Enop^{a,\\beta,c-1}$ such that $z_{a,\\beta,c}=\\partial_1w$. In this case, $z':=z-\\partial w$ has the same boundary as $z$, and the maximal $c'$ such that $z'_{a,b-1-c',c'}\\neq 0$ verifies $c'<c$. Thus, we can apply again the previous argument until we get an element of $\\Enop^{\\geq a,\\bul,\\bul}$, or until we are in the following case.\n\n\\item If $c=0$, then $z_{a,\\beta,0}\\in\\i(C^{a,\\beta})$.\n\\end{itemize}\n\nWe deduce that, up to a cycle, we can assume that $z_1=z_1'+z_1''$ with $z_1'\\in\\i(C^{a,\\bul})$ and $z_1''\\in\\Enop^{\\geq a+1,\\bul,\\bul}$, i.e., $z\\in\\i(C^{\\geq a-k+1,\\bul})+\\Enop^{\\geq a+1,\\bul,\\bul}$.\n\n\\medskip\n\nTherefore,\n\\begin{align*}\n  &\\hspace{-2cm}x \\in \\bigl(\\Enop^{\\geq a+1,\\bul,\\bul}+\\partial\\bigl(\\i C^{\\geq a-k+1,\\bul}+\\Enop^{\\geq a+1,\\bul,\\bul}\\bigr)\\bigr)\\cap\\Im(\\i) \\\\\n  &= \\bigl(\\Enop^{\\geq a+1,\\bul,\\bul} + \\i\\d C^{\\geq a-k+1,\\bul}\\bigr)\\cap\\Im(\\i) \\\\\n  &= \\Enop^{\\geq a+1,\\bul,\\bul}\\cap\\Im(\\i) + \\i\\d C^{\\geq a-k+1,\\bul} \\\\\n  &= \\i\\bigl(C^{\\geq a+1,\\bul} + \\d C^{\\geq a-k+1,\\bul}\\bigr).\n\\end{align*}\nWe can replace Equation \\eqref{eqn:EI2} by\n\\[ \\prescript{\\textup I}{}\\Enop_k^{a,b} = \\frac{\\Bigl\\{x\\in\\Tot^{a+b}(\\Enop^{\\bul,\\bul,\\bul})\\cap\\Enop^{\\geq a,\\bul,\\bul} \\st \\partial x \\in \\Enop^{\\geq a+k,\\bul,\\bul}\\Bigr\\} \\cap \\Im(\\i)}  {\\i\\bigl(C^{\\geq a+1,\\bul} + \\d C^{\\geq a-k+1,\\bul}\\bigr)}. \\]\nThe injectivity of $\\i\\colon \\prescript{\\textup I}{}\\Enop_k^{a,b}\\to\\prescript{\\textup I}{}C_k^{a,b}$ now follows from the injectivity of $\\i\\colon C^{\\bul,\\bul}\\to\\Enop^{\\bul,\\bul,0}$, which concludes the proof of the spectral resolution lemma.\n\\end{proof}\n}\n\n\\vspace{.5cm}\nThe two remaining section contain the proofs of the two Propositions~\\ref{prop:STinf_to_ST} and~\\ref{prop:app1}.\n\n\n\n\\subsection{Vanishing of the cohomology of $\\Tot^\\bul{\\STinf{p}^{\\sigma,\\bul,\\bul}}$ for $\\sigma\\neq\\conezero$} \\label{sec:STinf_to_ST}\n\nIn this section, we prove Proposition \\ref{prop:STinf_to_ST}, which, we recall, claims that the total cohomology of $\\STinf{p}^{\\sigma,\\bul,\\bul}_0$ is trivial for $\\sigma\\neq\\conezero$, where\n\\[ \\STinf{p}^{\\sigma,a,b}:=\n\\begin{cases}\n  \\bigoplus_{\\delta\\in X \\\\ \\maxsed(\\delta)=\\sigma \\\\ \\dims\\delta=p+a-b \\\\ \\dims{\\delta_\\infty}\\leq a} H^{2b}(\\delta) & \\text{if $b\\leq p$}, \\\\\n  0 & \\text{otherwise.}\n\\end{cases} \\]\nThe differential of bidegree $(1,0)$ is $\\i^*+\\pi^*$, and the differential of bidegree $(0,1)$ is $\\gys$.\n\nThis double complex can be unfolded into a triple complex: $\\Tot^\\bul(\\STinf{p}^{\\sigma,\\bul,\\bul})=\\Tot^\\bul(\\Enop^{\\bul,\\bul,\\bul})$ where\n\\[ \\Enop^{a,b,a'}:=\n\\begin{cases}\n  \\bigoplus_{\\delta\\in X \\\\ \\maxsed(\\delta)=\\sigma \\\\ \\dims\\delta=p+a+a'-b \\\\ \\dims{\\delta_\\infty}=a'} H^{2b}(\\delta) & \\text{if $b\\leq p$ and $a\\geq 0$}, \\\\\n  0 & \\text{otherwise.}\n\\end{cases} \\]\nwhere the differentials of multidegree $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ are respectively $\\i^*$, $\\gys$ and $\\pi^*$.\n\nTo prove that the total cohomology of $\\Enop^{\\bul,\\bul,\\bul}$ is trivial, it suffices to show that $(\\Enop^{a,b,\\bul}, \\pi^*)$ is exact for any integers $a$ and $b$.\n\n\\medskip\n\nLet $\\eta$ be a face of sedentarity $\\conezero$ such that $\\maxsed(\\eta)=\\sigma$. Notice that $\\sed(\\eta)=\\conezero$ implies $\\maxsed(\\eta)=\\eta_\\infty$. For any cone $\\tau\\prec\\sigma$, we denote by $\\eta_\\infty^\\tau$ the face of $\\eta$ of sedentarity $\\tau$, which is given by\n\\[ \\eta_\\infty^\\tau = \\eta\\cap N^\\tau_\\mathbb{R}. \\]\nNote that each face of $X$ of max-sedentarity $\\sigma$ can be written in a unique way in this form. Notice also that\n\\begin{itemize}\n\\item $\\pi_\\sigma$ induces an isomorphism from $\\eta_\\f$ to $(\\eta_\\infty^\\tau)_\\f$.\n\\item $\\Sigma^\\eta\\simeq\\Sigma^{\\eta_\\infty^\\tau}$. In particular $H^\\bul(\\eta_\\infty^\\tau)\\simeq H^\\bul(\\eta)$.\n\\item $\\pi^*_{\\delta\\ssupface\\gamma}$ is nontrivial if and only if $\\gamma=\\eta_\\infty^\\tau$ and $\\delta=\\eta_\\infty^\\zeta$ for some $\\eta$ of sedentarity $\\conezero$, and some pair of cones $\\zeta\\ssubface\\tau$ which are faces of $\\maxsed(\\eta)$.\n\\end{itemize}\n\n\\medskip\n\nFrom all these observations, we deduce that the sequence $(\\Enop^{a,b,\\bul},\\pi^*)$ can be decompose as a direct sum:\n{ \\renewcommand{\\S}[1]{\\bigoplus_{\\tau\\prec\\sigma \\\\ \\dims\\tau=#1}H^{2b}(\\eta_\\infty^\\tau)}\n\\[ \\bigoplus_{\\eta\\in X^\\conezero \\\\ \\maxsed(\\eta)=\\sigma \\\\ \\dims{\\eta}=p+a+\\dims\\sigma-b} \\Bigl( 0 \\to H^{2b}(\\eta_\\infty^\\sigma) \\to \\S{\\dims\\sigma-1} \\to \\dots \\to \\S{1} \\to H^{2b}(\\eta) \\to 0 \\Bigr). \\] }\nThis sequence can be rewritten as\n{ \\renewcommand{\\S}[1]{\\bigoplus_{\\tau\\prec\\sigma \\\\ \\dims\\tau=#1}\\mathbb{R}}\n\\[ \\bigoplus_{\\eta\\in X^\\conezero \\\\ \\maxsed(\\eta)=\\sigma \\\\ \\dims{\\eta}=p+a+\\dims\\sigma-b} \\Bigl( 0 \\to \\mathbb{R} \\to \\S{\\dims\\sigma-1} \\to \\dots \\to \\S{1} \\to \\mathbb{R} \\to 0 \\Bigr) \\otimes H^{2b}(\\eta). \\] }\n\nThus, it just remains to prove that these sequences are exact for every $\\eta$. The complex is clearly isomorphic to the simplicial homology of $\\sigma$ with real coefficients. This homology is trivial since it corresponds to the reduced homology of a simplex.\n\n\\medskip\n\nThis concludes the proof of Proposition \\ref{prop:STinf_to_ST} and thus that of Theorem \\ref{thm:steenbrink}. \\QED\n\n\n\n\\subsection{Proof of Proposition~\\ref{prop:app1}}\n\\label{sec:proofapp1}\n\nIn this final part of this section, we prove Proposition~\\ref{prop:app1} which concerns the three basic relations concerning the composition of maps $\\gys$ and $\\i^*$.\n\n\\medskip\n\nLet us start by proving that the composition $\\gys\\circ\\gys$ is vanishing.\n\nLet $a,b$ be a pair of integers, and let $s\\geq\\abs{a}$ be an integer with $s\\equiv a\\mod2$. Let $x\\in\\ST_1^{a,b,s}$. We have to prove that $\\gys\\circ\\gys(x)=0$. By linearity of the Gysin maps, we can assume that $x\\in H^{a+b-s}(\\delta)$ for some $\\delta\\in X_\\f$ of dimension $s$. If $s-1<\\abs{a+1}$, or $s-2<\\abs{a+2}$, then $\\gys\\circ\\gys(x)=0$. Otherwise,\n\\[ \\gys\\circ\\gys(x)=\\sum_{\\gamma\\ssubface\\delta}\\sum_{\\nu\\ssubface\\gamma}\\sign(\\nu, \\gamma)\\sign(\\gamma, \\delta)\\gys_{\\gamma\\ssupface\\nu}\\circ\\gys_{\\delta\\ssupface\\gamma}(x). \\]\nThis sum can be rewritten in the form\n\\[ \\gys\\circ\\gys(x)=\\sum_{\\nu\\prec\\delta \\\\ \\dims{\\nu}=s-2}\\sum_{\\gamma\\ssubface\\delta \\\\ \\gamma\\ssupface\\nu}\\sign(\\nu, \\gamma)\\sign(\\gamma, \\delta)\\gys_{\\gamma\\ssupface\\nu}\\circ\\gys_{\\delta\\ssupface\\gamma}(x). \\]\nIf $\\nu$ is a subface of $\\delta$ of codimension 2, then, by the diamond property, there exist exactly two faces $\\gamma$ and $\\gamma'$ such that $\\nu\\ssubface\\gamma,\\!\\gamma'\\ssubface\\delta$. Moreover, the composition is just the natural Gysin map from $\\delta$ to $\\nu$. Thus, we have\n\\[ \\gys_{\\gamma'\\ssupface\\nu}\\circ\\gys_{\\delta\\ssupface\\gamma'}(x)=\\gys_{\\gamma\\ssupface\\nu}\\circ\\gys_{\\delta\\ssupface\\gamma}(x), \\]\nand\n\\[ \\sign(\\nu, \\gamma)\\sign(\\gamma, \\delta)=-\\sign(\\nu, \\gamma')\\sign(\\gamma', \\delta). \\]\nThis shows that the previous sum vanishes, which proves $\\gys\\circ\\gys=0$.\n\n\\medskip\n\nOne can prove in the same way that $\\i^*\\circ \\i^*=0$, so we omit the details.\n\n\\medskip\n\nWe now study the map $\\i^*\\circ\\gys+\\gys\\circ \\i^*$. Let $a, b, s, \\delta$ and $x\\in H^{a+b-s}(\\delta)\\subseteq\\ST_1^{a,b}$ be defined as above. If $s-1<\\abs{a+1}$, then $a\\geq0$ and $s<\\abs{a+2}$ and the codomain of $\\i^*\\circ\\gys+\\gys\\circ \\i^*$ is trivial. Otherwise, as in the previous paragraph, we can decompose $\\i^*\\circ\\gys(x)+\\gys\\circ \\i^*(x)$ as a sum\n\\[ \\sum_{\\delta'\\in X_\\f \\\\ \\dims{\\delta'}=s} \\Bigl(S_{\\prec,\\delta'}+S_{\\succ,\\delta'} \\Bigr)\\]\nwhere\n\\begin{gather*}\nS_{\\prec,\\delta'} =\n  \\sum_{\\gamma\\ssubface\\delta \\\\ \\gamma\\ssubface\\delta'}\n    \\sign(\\gamma,\\delta')\\sign(\\gamma,\\delta)\\i^*_{\\gamma\\ssubface\\delta'}\\circ\\gys_{\\delta\\ssupface\\gamma}(x)\\text{ and } \\\\\nS_{\\succ,\\delta'} =\n  \\sum_{\\eta\\ssupface\\delta \\\\ \\eta\\ssupface\\delta'}\n    \\sign(\\delta',\\eta)\\sign(\\delta,\\eta)\\gys_{\\eta\\ssupface\\delta'}\\circ \\i^*_{\\delta\\ssubface\\eta}(x),\n\\end{gather*}\nand the sums are over faces in $X_\\f$. We claim that for each $\\delta'$, the sum $S_{\\prec,\\delta'}+S_{\\succ,\\delta'}$ is zero. This will finish the proof of our proposition.\n\n\\medskip\n\n\\noindent Four cases can happen.\n\n\\begin{enumerate}[label={\\textnormal{(\\textit{\\roman*})}}, leftmargin=0pt]\n\\item \\label{enum:gys_i*:1} Suppose $\\delta'\\neq\\delta$ and $\\dims{\\delta\\cap\\delta'}<s-1$.\n\\end{enumerate}\nIn this case, both sums $S_{\\prec,\\delta'}$ and $S_{\\succ,\\delta'}$ have no terms, and they are both zero.\n\n\\medskip\n\n\\begin{enumerate}[resume*]\n\\item \\label{enum:gys_i*:2} Suppose $\\delta'\\neq\\delta$, the face $\\gamma:=\\delta\\cap\\delta'$ is of dimension $s-1$, and no face $\\eta$ of dimension $s+1$ contains both $\\delta$ and $\\delta'$.\n\\end{enumerate}\n\nIn this case, $S_{\\succ,\\delta'}$ is a sum with no term. The other sum, $S_{\\prec,\\delta'}$, contains only one term which is $\\i^*_{\\gamma\\ssubface\\delta'}\\circ\\gys_{\\delta\\ssupface\\gamma}(x)$. This term is zero because, the ray corresponding to $\\delta$ and the ray corresponding to $\\delta'$ in the fan $\\Sigma^\\gamma$, are not comparable.\n\n\\begin{enumerate}[resume*]\n\\item \\label{enum:gys_i*:3} Suppose $\\delta'\\neq\\delta$, the face $\\gamma:=\\delta\\cap\\delta'$ is of dimension $s-1$, and there exists a face $\\eta\\in X$ of dimension $s+1$ containing both $\\delta$ and $\\delta'$.\n\\end{enumerate}\n\nIn this case, notice that $\\eta=\\conv(\\delta\\cup\\delta')$, thus $\\eta\\in X_\\f$. Moreover the four faces form a diamond, i.e., we have $\\gamma\\ssubface\\, \\delta,\\delta'\\ssubface\\eta$. We infer that both sums $S_{\\ssubface,\\delta'}$ and $S_{\\ssupface,\\delta'}$ contain a unique term, and the sum is given by\n\\[ \\sign(\\gamma,\\delta')\\sign(\\gamma,\\delta)\\i^*_{\\gamma\\ssubface\\delta'}\\circ\\gys_{\\delta\\ssupface\\gamma}(x)\n+ \\sign(\\delta',\\eta)\\sign(\\delta,\\eta)\\gys_{\\eta\\ssupface\\delta'}\\circ \\i^*_{\\delta\\ssubface\\eta}(x). \\]\nThis is zero because\n\\begin{gather*}\n\\i^*_{\\gamma\\ssubface\\delta'}\\circ\\gys_{\\delta\\ssupface\\gamma}(x)=\\gys_{\\eta\\ssupface\\delta'}\\circ \\i^*_{\\delta\\ssubface\\eta}(x), \\text{ and } \\\\\n\\sign(\\gamma,\\delta')\\sign(\\gamma,\\delta)=-\\sign(\\delta',\\eta)\\sign(\\delta,\\eta).\n\\end{gather*}\n\n\\medskip\n\nIt remains only the case $\\delta =\\delta'$.\n\n\\begin{enumerate}[resume*]\n\\item \\label{enum:gys_i*:4} Suppose $\\delta=\\delta'$.\n\\end{enumerate}\n\nThis case is more technical. We need some notations. Let $v_0, \\dots, v_s$ be the vertices of $\\delta$. For $j\\in\\zint0s$ let $\\gamma_j\\ssubface\\delta$ be the face opposite to $v_j$. Let $\\eta_1, \\dots, \\eta_r$ be the compact faces of dimension $s+1$ containing $\\delta$ and $\\eta_{r+1}, \\dots, \\eta_{r+t}$ be the non-compact ones. For each $i\\in\\zint1r$ we denote by $w_i$ the only vertex of $\\eta_i$ which is not in $\\delta$. For each $i\\in\\zint{r+1}{r+t}$, we define $u_i$ to be the primitive vector of the ray $\\eta_{i,\\infty}$. We will work with the Chow rings. For each $i\\in\\zint1{r+t}$, the face $\\eta_i$ corresponds to a ray $\\rho_i$ in $\\Sigma^\\delta$ and to an element $x_i$ in $A^1(\\delta)$. Note that every ray of $\\Sigma^\\delta$ is of this form. Moreover, for any $j\\in\\zint0s$, each ray $\\rho_i$ corresponds to a ray in $\\Sigma^{\\gamma_j}$ which we will also denote by $\\rho_i$. We also denote by $x_i$ the corresponding element in $A^1(\\gamma_j)$. We denote by $\\rho_{\\delta,j}$ the ray of $\\Sigma^{\\gamma_j}$ corresponding to $\\delta$, and by $x_{\\delta,j}$ the associated element in $A^1(\\gamma_j)$. If $\\ell$ is a linear form on $N^\\eta_\\mathbb{R}$ for some face $\\eta$, and if $\\varrho$ is a ray in $\\Sigma^\\eta$, we will write $\\ell(\\varrho)$ for the value of $\\ell$ on the primitive vector of $\\varrho$.\n\n\\medskip\n\nSince $\\gys$ and $\\i^*$ are ring homomorphisms, we can assume without loss of generality that the chosen element in $\\ST_1^{a,b,s}(X)$ is the unity $1$ of $A^0(\\Sigma)$. Then, for any $i\\in\\zint1r$,\n\\[ \\gys_{\\eta_i\\ssupface\\delta}\\circ\\i^*_{\\delta\\ssubface\\eta_i}(1)=x_i. \\]\n\n\\medskip\n\nIn this case, $S_{\\succ,\\delta}$ is given by the sum over \\emph{compact} faces $\\eta$ of dimension $s+1$ containing $\\delta$ of\n\\[ \\sign(\\delta,\\eta)^2\\gys_{\\eta\\ssupface\\delta}\\circ\\i^*_{\\delta\\ssubface\\eta}(1). \\]\nIt follows that\n\\[ S_{\\succ,\\delta}=\\sum_{i=1}^rx_i. \\]\n\n\\medskip\n\nLet $j\\in\\zint0s$. We have $\\gys_{\\delta\\ssupface\\gamma_j}(1)=x_{\\delta,j}\\in A^1(\\gamma_j)$. Choose an affine form $\\ell_j$ on $N_\\mathbb{R}$ that is zero on $\\gamma_j$ and $-1$ on $v_j$. Let $\\ell^0_j$ be the corresponding linear form. It induces a linear form $\\lambda_j$ on $N^{\\gamma_j}_\\mathbb{R}$. Moreover, we have the following properties for $\\lambda_j$:\n\\begin{gather*}\n\\lambda_j(\\rho_{\\delta,j})=\\ell(v_j)=-1, \\\\\n\\forall i\\in\\zint1r, \\,\\, \\lambda_j(\\rho_i)=\\ell_j(w_i), \\, \\text{ and } \\\\\n\\forall i\\in\\zint{r+1}{r+t},\\,\\, \\lambda_j(\\rho_i)=\\ell^0_j(u_i).\n\\end{gather*}\nThus, $x_{\\delta,j}\\in A^1(\\gamma_j)$ is equal to\n\\[ \\sum_{i=1}^r\\ell_j(w_i)x_i+\\sum_{i=r+1}^{r+t}\\ell^0_j(u_i)x_i+x', \\]\nwhere $x'\\in A^1(\\gamma_j)$ is incomparable with $x_{\\delta,j}$ in the sense that $x'x_{\\delta,j}=0$. Hence, the image of $x_{\\delta,j}$ by $\\i^*_{\\gamma_j\\ssubface\\delta}$ is\n\\[ \\i^*_{\\gamma_j\\ssubface\\delta}\\circ\\gys_{\\delta\\ssupface\\gamma_j}(1)= \\sum_{i=1}^r\\ell_j(w_i)x_i+\\sum_{i=r+1}^{r+t}\\ell^0_j(u_i)x_i\\in A^1(\\delta). \\]\nSet $\\ell=\\ell_0+\\dots+\\ell_s$ and denote by $\\ell^0$ the corresponding linear form. Summing over all $\\gamma_j$, we get\n\\[ S_{\\prec, \\delta}=\\sum_{i=1}^r\\ell(w_i)x_i+\\sum_{i=r+1}^{r+t}\\ell^0(u_i)x_i. \\]\nLet $\\cteun$ be the constant function on $N_\\mathbb{R}$ which takes value $1$ everywhere. We can rewrite\n\\[ S_{\\succ, \\delta}=\\sum_{i=1}^rx_i=\\sum_{i=1}^r\\cteun(w_i) x_i. \\]\n\n\\medskip\n\nSet $\\ell'=\\ell+\\cteun$ and let $\\ell^{\\prime 0}=\\ell^0$ be the corresponding linear form. We have\n\\[ S_{\\prec, \\delta}+S_{\\succ, \\delta}=\\sum_{i=1}^r\\ell'(w_i)x_i+\\sum_{i=r+1}^{r+t}\\ell^{\\prime 0}(u_i)x_i. \\]\nSince $\\ell$ equals $-1$ on each vertex of $\\delta$, $\\ell'$ is zero on $\\delta$. Thus, it induces a linear form $\\lambda'$ on $N^\\delta_R$ which verifies\n\\begin{gather*}\n\\forall i\\in\\zint1r, \\lambda'(\\rho_i)=\\ell'(w_i)\\text{ and } \\\\\n\\forall i\\in\\zint{r+1}{r+t}, \\lambda'(\\rho_i)=\\ell^{\\prime 0}(u_i).\n\\end{gather*}\nTherefore, $S_{\\prec, \\delta}+S_{\\succ, \\delta}=0$ in $A^1(\\delta)$, which is what we wanted to prove.\n\n\\medskip\n\nIn each of the four cases above we showed that $S_{\\prec, \\delta'}+S_{\\succ, \\delta'}=0$, thus $\\gys\\circ \\i^*(x)+\\i^*\\circ\\gys(x)=0$.\n\n\\medskip\n\nThe proof of Proposition~\\ref{prop:app1} is complete.\n\n\n\n\n\n\n\\section{K\\\"ahler tropical varieties} \\label{sec:kahler}\nThe aim of this section is to show that $\\ST_1^{\\bul, \\bul}$ and the tropical cohomology $H^{\\bul,\\bul}_\\trop$ admit a Hodge-Lefschetz structure. The precise meaning will only appear in Section~\\ref{sec:differential_HL_structure} where we give a precise definition of Hodge-Lefschetz structures and treat the consequences of the materials introduced in this section in detail.\n\n\\medskip\n\nWe introduce the monodromy and Lefschetz operators on $\\ST_1^{\\bul, \\bul}$ in this section. The definition of the latter is based on the definition of K\\\"ahler forms in tropical geometry which are introduced in this section as well.\n\n\n\n\\subsection{Monodromy operator $N$ on $\\ST_1^{\\bul, \\bul}$} The monodromy operator is the data of maps $N^{a,b} \\colon \\ST_1^{a,b} \\to \\ST_1^{a+2, b-2}$ defined as follows. Writing\n$\\ST_1^{a,b} = \\bigoplus \\ST_1^{a,b,s}$ for $s \\geq \\abs{a}$ with $s \\equiv a (\\mod 2)$, then $N^{a,b}$ is defined by setting its restrictions $N^{a,b,s}$ on\n$\\ST_1^{a,b,s}$ to be\n\n\\begin{equation}\nN^{a,b, s} = \\begin{cases} \\mathrm{id} \\colon\\ST_1^{a,b,s} \\to \\ST_1^{a+2,b-2,s}   & \\textrm{if } s \\geq \\abs{a+2}, \\\\\n0 & \\textrm{otherwise}.\n\\end{cases}\n\\end{equation}\n\nIf there is no risk of confusion, we drop all the superscripts and denote simply by $N$ the operator $N^{a, b}$ and all its restrictions $N^{a,b,s}$.\n\\begin{remark} More precisely, we could view the collection of maps $N^{a,b}$ as a map $N$ of bidegree $(2, -2)$\n\\[N = \\bigoplus_{a,b}N^{a,b} \\colon \\bigoplus_{a,b} \\ST_1^{a,b} \\longrightarrow \\bigoplus_{a,b} \\ST_1^{a,b}.\\]\n\\end{remark}\nWe have the following proposition which summaries the main properties of the monodromy operator.\n\n\\begin{prop} \\label{prop:N_commutes} We have\n\\begin{itemize}\n\\item $[N,\\i^*] = [N, \\gys] =0$\n\\item for $a\\leq 0$, we have an isomorphism $N^{-a} \\colon \\ST_1^{a,b} \\to \\ST_1^{-a, b+2a} $\n\\item for $a \\leq 0$, $\\ker N^{-a+1} \\cap \\ST_1^{a,b} = \\ST_1^{a, b, -a}$.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nFor the first point, let $a, b, s$ be three integers with $a \\equiv s \\pmod 2$. We have the following commutative diagrams.\n\\[ \\begin{tikzcd}\n\\ST^{a,b,s}_1 \\rar[\"\\id\"]\\dar[\"\\i^*\"] & \\ST^{a+2,b-2,s}_1 \\dar[\"\\i^*\"] \\\\\n\\ST^{a+1,b,s+1}_1 \\rar[\"\\id\"] & \\ST^{a+3,b-2,s+1}_1\n\\end{tikzcd} \\qquad \\begin{tikzcd}\n\\ST^{a,b,s}_1 \\rar[\"\\id\"]\\dar[\"\\gys\"] & \\ST^{a+2,b-2,s}_1 \\dar[\"\\gys\"] \\\\\n\\ST^{a,b+2,s}_1 \\rar[\"\\id\"] & \\ST^{a+2,b,s}_1\n\\end{tikzcd} \\]\nTo get the commutativity in $\\ST_1^{\\bul,\\bul}$, it remains to check that, in each diagram, if the top-left and bottom-right pieces appear in $\\ST_1^{\\bul,\\bul}$, then the two other pieces appear as well. This comes from the following facts, which are easy to check.\n\\begin{gather*}\ns \\equiv a \\equiv a+2 \\pmod 2 \\quad\\text{ and }\\quad s+1 \\equiv a+1 \\equiv a+3 \\pmod 2, \\text{ and } \\\\\ns\\geq\\abs{a} \\text{ and } s+1\\geq\\abs{a+3} \\Longrightarrow s\\geq\\abs{a+2} \\text{ and } s+1\\geq\\abs{a+1}.\n\\end{gather*}\n\n\\medskip\n\nFor the second point, for $a\\leq0$, $s\\geq\\abs{a}$ implies $s\\geq\\abs{a+2k}$ for any $k\\in\\zint0{-a}$. Thus $N^{-a}$ induces the isomorphism $\\ST_1^{a,b,s}\\simeq\\ST_1^{-a,b+2a,s}$, which implies the result.\n\n\\medskip\n\nFor the last point, for $a\\leq0$, we have\n\\begin{align*}\n\\ker N^{-a+1}\\cap\\ST_1^{a,b}\n  &= (N^{-a})^{-1}\\ker\\bigl(N\\colon\\ST_1^{-a,b+2a}\\to\\ST_1^{-a+2,b+2a-2}\\bigr) \\\\\n  &= (N^{-a})^{-1}\\ST_1^{-a,b+2a,-a} \\\\\n  &= \\ST_1^{a,b,-a}. \\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{defi}[Primitive parts of the monodromy]\nFor $a\\leq 0$, the kernel of $N^{-a+1}$ on $\\ST_1^{a,b}$ is called the \\emph{primitive part of bidegree $(a,b)$} for the monodromy operator $N$.\n\\end{defi}\n\n\n\n\\subsection{K\\\"ahler forms}\n\n\n\\subsubsection{Local ample classes} Let $\\Sigma\\subseteq N_\\mathbb{R}$ be a (unimodular) Bergman fan. Recall that an element $\\ell\\in A^1(\\Sigma)$ is called an ample class if it corresponds to a strictly convex cone-wise linear function on $\\Sigma$.\n\n\\begin{prop} \\label{prop:ample_stable}\nLet $\\Sigma$ be a Bergman fan. Let $\\ell\\in A^1(\\Sigma)$ be an ample class of $\\Sigma$. Then\n\\begin{itemize}\n\\item $\\ell$ verifies the Hodge-Riemann relations $\\HR(\\Sigma,\\ell)$,\n\\item for any cone $\\sigma\\in\\Sigma$, $\\i^*_\\sigma(\\ell)\\in A^1(\\Sigma^\\sigma)$ is an ample class of $\\Sigma^\\sigma$.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nThe first point is Theorem \\ref{thm:mainlocal}. The second point follows from Proposition~\\ref{prop:ell_f_independent} combined with \\ref{thm:mainlocal}.\n\\end{proof}\n\n\n\\subsubsection{K\u00e4hler class} Let $X$ be a unimodular polyhedral complex structure on a smooth compact tropical variety.\n\n\\begin{defi}\nA \\emph{K\u00e4hler class} for $X$ is the data of ample classes $\\ell^v \\in H^2(v)$ for each vertex $v\\in X_\\f$ such that, for each edge $e$ of $X_\\f$ of extremities $u$ and $v$, we have the compatibility\n\\[\\i_{u \\ssubface e}^*(\\ell^u) = \\i_{v \\ssubface e}^*(\\ell^v)\\]\nin $H^2(e)$.\n\\end{defi}\n\n\\begin{thm}\\label{thm:KahlerClass}\nEach K\u00e4hler class defines an element $\\omega \\in H^{1,1}(X)$.\n\\end{thm}\n\n\\begin{proof}\nLet $\\ell$ be a K\u00e4hler class. We have\n\\[ \\ell \\in \\bigoplus_{v\\in X_\\f \\\\ \\dims{v}=0}H^2(v) = \\ST_1^{0,2,0}\\subseteq \\ST_1^{0,2}. \\]\nThus, $\\ell$ can be seen as an element of $\\ST_1^{0,2}$. Moreover,\n\\begin{align*}\n\\d\\ell\n  &= \\sum_{v\\in X_\\f \\\\ \\dims{v}=0}\\sum_{e\\in X_\\f \\\\ e\\ssupface v}\\sign(v,e)\\i^*_{v\\ssubface e}(\\ell^v) \\\\\n  &= \\sum_{e\\in X_\\f \\\\ \\dims{e}=1 \\\\ u,v\\ssubface e}\\sign(u,e)\\i^*_{u\\ssubface e}(\\ell^u)+\\sign(v,e)\\i^*_{v\\ssubface e}(\\ell^v), \\\\\n&= 0.\n\\end{align*}\nHence, $\\ell$ is in the kernel of $\\ST_1^{0,2}$ and thus induces an element $\\omega\\in \\ST_2^{0,2}\\simeq H^{1,1}(X)$.\n\\end{proof}\n\n\\begin{defi} \\begin{itemize}[label=-]\n\\item An element $\\omega$ in $H^{1,1}(X)$ is called a \\emph{K\u00e4hler class} if there exist ample elements $\\ell^v \\in H^2(v)$ as above which define $\\omega \\in H^{1,1}(X)$.\n\\item A smooth compact tropical variety is called \\emph{K\\\"ahler} if it admits a unimodular triangulation $X$ and a K\u00e4hler class $\\omega\\in H^{1,1}(X)$.\n\\end{itemize}\n\n\\end{defi}\n\n\n\\subsubsection{Projective tropical varieties are K\\\"ahler}\n\\label{sec:projective_Kahler}\n\nIn this section, we consider the case of a unimodular polyhedral complex $Y \\subseteq \\mathbb{R}^n$ and the compactification $X$ of $Y$ in the tropical toric variety $\\TP_{Y_\\infty}.$ In this situation $Y\\subseteq\\mathbb{R}^n$ is the part of $X$ of sedentarity $\\conezero$.\n\nWe assume that the triangulation $Y$ is quasi-projective. Under this assumption, we will show that $X$ is K\u00e4hler, i.e., that there exists a K\u00e4hler class in $H^{1,1}(X)$.\n\n\\medskip\n\nLet $f$ be a strictly convex piecewise linear function on $Y$. Let $\\delta$ be a face of $X_\\f$, and $\\phi$ be an affine linear form with the same restriction to $\\delta$ as $f$. Denote by $f^\\delta$ the cone-wise linear function induced by $f-\\phi$ on $\\Sigma^\\delta$.\n\n\\begin{propdefi} \\label{propdefi:convex_subface}\nThe function $f^\\delta$ is strictly convex. Denote by $\\ell^\\delta$ the corresponding element $\\ell(f^\\delta)$ of $H^2(\\delta)$. This element $\\ell^\\delta$ does not depend on the chosen $\\phi$. Moreover, if $\\gamma$ is a face of $\\delta$, then $\\ell^\\delta=\\i^*_{\\gamma\\prec\\delta}(\\ell^\\gamma)$.\n\\end{propdefi}\n\n\\begin{proof}\nFor both parts, one can adapt the proof given in Proposition~\\ref{prop:ell_f_independent} in the local case.\n\\end{proof}\n\nIt follows that $\\ell^\\delta$ is an ample class in $A^1(\\delta) = H^2(\\delta)$. In particular, this holds for vertices of $X_\\f$, giving an element $(\\ell^v)_{v \\text{ vertex of } X_\\f}\\in\\ST^{0,2}_1$. We get the following theorem.\n\n\\begin{thm} \\label{thm:projective_Kahler}\nA strictly convex function $f$ on $Y$ defines a K\u00e4hler class of $X$.\n\\end{thm}\n\\begin{proof}\nWe just have to notice that, for any edge $e\\in X_\\f$ of extremities $u$ and $v$,\n\\[ \\i^*_{v\\ssubface e}(\\ell^v)=\\ell^e=\\i^*_{u\\ssubface e}(\\ell^u). \\qedhere \\]\n\\end{proof}\n\n\\medskip\n\nThe two following results give a better understanding of the link between piecewise linear functions and elements of $H^{1,1}(\\mathfrak X)$. The corollary will be useful in Section \\ref{sec:proofmaintheorem}.\n\nLet $v$ be a vertex of $Y$. We define $\\chi_v$ the \\emph{characteristic function of $v$} to be the unique piecewise linear function on $Y$ which takes value $1$ at $v$, takes value $0$ at any other vertex of $Y$, and whose slope along any non-compact one dimensional face of $Y$ is zero.\n\n\\begin{prop}\nLet $v$ be a vertex of $Y$ and let $\\chi_v$ be the characteristic function of $v$. Then the class given by $\\chi_v$ in $H^{1,1}(\\mathfrak X)$ is trivial.\n\\end{prop}\n\n\\begin{proof}\nLet us also denote by $\\chi_v$ the corresponding element of $\\ST^{0,2}_1$. Then\n\\[ \\chi_v = -\\d\\,\\sum_{e \\in Y_\\f \\\\ e \\ssupface v}\\sign(v,e)1_e \\]\nwhere $1_e$ is the natural generator of $H^0(e)\\subseteq\\ST^{-1,2}$. Hence $\\chi_v$ is a boundary, and its cohomological class is trivial.\n\\end{proof}\n\nRecall that if $f$ is a piecewise linear function on $Y$, then we defined the asymptotic part $f_\\infty\\colon Y_\\infty \\to \\mathbb{R}$ of $f$ in Section \\ref{sec:X_infty_X}. We immediately get the following corollary.\n\n\\begin{cor} \\label{cor:class_independent_finite_part}\nLet $f$ be a piecewise linear function on $Y$ and let $\\ell$ be the corresponding class in $H^{1,1}(X)$. Assume moreover that $f_\\infty$ is well-defined. Then $\\ell$ only depends on $f_\\infty$.\n\\end{cor}\n\nAnother equivalent description of the class $\\ell$ is given in Section \\ref{sec:cl_D'}.\n\n\n\n\\subsection{The Lefschetz operator} Let $\\ell$ be a K\u00e4hler class of $X$. For each face $\\delta\\in X_\\f$, we define $\\ell^\\delta\\in H^2(\\delta)\\simeq A^1(\\delta)$ by\n\\[ \\ell^\\delta=\\i^*_{v\\prec\\delta}(\\ell^v), \\]\nwhere $v$ is any vertex of $\\delta$. This definition does not depend on the choice of $v$ because, if $u$ is another vertex of $\\delta$, we have\n\\[ \\i^*_{v\\prec\\delta}(\\ell^v)=\\i^*_{e\\prec\\delta}\\circ \\i^*_{v\\prec e}(\\ell^v)=\\i^*_{e\\prec\\delta}\\circ \\i^*_{u\\prec e}(\\ell^u)=\\i^*_{u\\prec\\delta}(\\ell^u), \\]\nwhere $e$ is the edge of extremities $u$ and $v$.\n\nLet $Q_\\delta$ be the bilinear form defined on $H^{2k}(\\delta) = A^k(\\delta)$, for $k \\leq \\frac{d-\\dims\\delta}2$, by\n\\[ \\forall \\, a,b\\in H^{2k}(\\delta), \\qquad Q_\\delta (a,b) = \\deg(ab(\\ell^\\delta)^{d-\\dims\\delta-2k}).\\]\nEvery $\\ell^\\delta$ is ample by Proposition \\ref{prop:ample_stable}.\n\nFor each $\\delta$, let\n\\[P^{2k}(\\delta) := \\ker\\Bigl((\\ell^\\delta)^{d-\\dims{\\delta}-2k+1} \\colon H^{2k}(\\delta) \\to H^{2d-2\\dims{\\delta}-2k+2}(\\delta)\\Bigr)\\]\nbe the primitive part of $H^{2k}(\\delta).$ Note that with the notations of Section~\\ref{sec:local}, we have $P^{2k}(\\delta) = A^k_{\\prim, \\ell^\\delta}(\\Sigma^\\delta)$.\n\n\\medskip\n\nThe following properties hold.\n\n\\begin{itemize}\n\\item (Hard Lefschetz) For each $k \\leq \\frac{d-\\dims{\\delta}}2$, the map\n\\[(\\ell^\\delta)^{d-\\dims\\delta -2k} \\colon H^{2k}(\\delta) \\to H^{2d-2\\dims{\\delta} -2k}(\\delta)\\]\nis an isomorphism.\n\\item (Hodge-Riemann) The bilinear form $(-1)^k Q_\\delta(\\,\\cdot\\,,\\cdot\\,)$ restricted to the primitive part $P^{2k}(\\delta) = \\ker\\Bigl((\\ell^\\delta)^{d-\\dims{\\delta}-2k+1} \\colon H^{2k}(\\delta) \\to H^{2d-2\\dims{\\delta}-2k+2}(\\delta)\\Bigr)$ is positive definite.\n\\item (Lefschetz decomposition) For $k\\leq\\frac{d-\\dims\\delta}2$, we have the orthogonal decomposition\n\\[ H^{2k}(\\delta)=\\bigoplus_{i=0}^k \\ell^{k-i}P^{2i}(\\delta), \\]\nwhere, for each $i\\in\\zint0k$, the map $\\ell^{k-i}\\colon P^{2i}(\\delta)\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} \\ell^{k-i}P^{2i}(\\delta)$ is an isomorphism preserving $Q_\\delta$.\n\\end{itemize}\n\n\\medskip\n\nWe now define the Lefschetz operator $\\ell^{a,b} \\colon \\ST_1^{a,b} \\to \\ST_1^{a, b+2} $ by\n\\[\\ell^{a,b} = \\bigoplus \\ell^{a,b,s} \\colon \\bigoplus_{s \\geq \\abs{a} \\\\\ns \\equiv a \\pmod 2} \\ST_1^{a,b, s} \\to \\bigoplus_{s \\geq \\abs{a} \\\\\ns \\equiv a \\pmod 2} \\ST_1^{a,b+2, s}\\]\nand the operator $\\ell^{a,b,s}$ is defined by\n\\[\\ell^{a,b,s} = \\bigoplus \\ell^\\delta \\colon \\bigoplus_{\\delta \\in X_\\f \\\\ \\dims{\\delta} =s} H^{a+b-s}(\\delta) \\longrightarrow    \\bigoplus_{\\delta \\in X_\\f \\\\ \\dims{\\delta} =s} H^{a+b+2-s}(\\delta).\\]\n\n\\medskip\n\nWe usually drop the indices $a,b$ and simply denote by $\\ell$ the operator of bidegree $(0,2)$.\n\n\\begin{remark} More precisely, we could view the collection of maps $\\ell^{a,b}$ as a map $\\ell$\n\\[\\ell = \\bigoplus_{a,b}\\ell^{a,b} \\colon \\bigoplus_{a,b} \\ST_1^{a,b} \\longrightarrow \\bigoplus_{a,b} \\ST_1^{a,b+2}.\\]\n\\end{remark}\n\nWe now recast Lemma~\\ref{lem:i_gys_basic_properties-local} in the following.\n\n\\begin{prop} \\label{lem:i_gys_basic_properties}\nLet $\\gamma\\ssubface\\delta$ be a pair of faces, and let $x\\in A^\\bul(\\gamma)$ and $y\\in A^\\bul(\\delta)$. Denote by $\\rho_{\\delta\/\\gamma}$ the ray associated to $\\delta$ in $\\Sigma^\\gamma$, and by $x_{\\delta\/\\gamma}$ the associated element of $A^1(\\gamma)$.\n\\begin{gather}\n\\text{$\\i^*_{\\gamma\\ssubface\\delta}$ is a surjective ring homeomorphism,} \\label{eqn:i_surjective_homeo} \\\\\n\\gys_{\\delta\\ssupface\\gamma}\\circ\\i^*_{\\gamma\\ssubface \\delta}(x)=x_{\\delta\/\\gamma}\\cdot x, \\label{eqn:gys_circ_i} \\\\\n\\gys_{\\delta\\ssupface\\gamma}(\\i^*_{\\gamma\\ssubface\\delta}(x)\\cdot y)=x\\cdot\\gys_{\\delta\\ssupface\\gamma}(y). \\label{eqn:gys_i_simplification}\n\\end{gather}\nMoreover, if $\\deg_\\gamma\\colon A^{d-\\dims\\gamma}\\to\\mathbb{R}$ and $\\deg_\\delta\\colon A^{d-\\dims\\delta}\\to\\mathbb{R}$ are the natural degrees map, then\n\\begin{equation} \\label{eqn:deg_circ_gys}\n\\deg_\\delta=\\deg_\\gamma\\circ\\gys_{\\delta\\ssupface\\gamma}.\n\\end{equation}\nFinally, $\\gys_{\\delta\\ssupface\\gamma}$ and $\\i^*_{\\gamma\\ssubface\\delta}$ are dual in the sense that\n\\begin{equation} \\label{eqn:i_gys_dual}\n\\deg_\\gamma(x\\cdot\\gys_{\\delta\\ssupface\\gamma}(y))=\\deg_\\delta(\\i^*_{\\gamma\\ssubface\\delta}(x)\\cdot y).\n\\end{equation}\n\\end{prop}\n\n\\begin{proof} The proof is similar to the one given in the local case.\n\\end{proof}\n\n\\medskip\n\nThe following proposition summarizes basic properties of the Lefschetz operator $\\ell$.\n\\begin{prop} We have\n\\begin{itemize}\n\\item $[\\ell, N] = 0$, $[\\ell, \\i^*] = [\\ell, \\gys] =0$.\n\\item for a pair of integers $a,b$ with $a+b \\geq d$, the map $\\ell^{d-a-b} \\colon \\ST_1^{a,b} \\longrightarrow \\ST_1^{a, 2d - 2a-b}$ is an isomorphism.\n\\end{itemize}\n\\end{prop}\n\\begin{proof}\nFor the first point, let $\\gamma\\ssubface\\delta$ be a pair of faces of $X_\\f$. We recall that $\\i^*_{\\gamma\\ssubface\\delta}\\colon A^\\bul(\\gamma)\\to A^\\bul(\\delta)$ is a surjective ring homeomorphism. Thus, for any $a\\in A^\\bul(\\gamma)$,\n\\[ \\i^*_{\\gamma\\ssubface\\delta}(\\ell^\\gamma a)=\\i^*_{\\gamma\\ssubface\\delta}(\\ell^\\gamma)\\i^*_{\\gamma\\ssubface\\delta}(a)=\\ell^\\delta\\i^*_{\\gamma\\ssubface\\delta}(a). \\]\nBy Equation \\eqref{eqn:gys_i_simplification}, for any $x\\in A^\\bul(\\delta)$,\n\\[ \\gys_{\\delta\\ssupface\\gamma}(\\ell^\\delta x)=\\gys_{\\delta\\ssupface\\gamma}(\\i^*_{\\gamma\\ssubface\\delta}(\\ell^\\gamma) x)=\\ell^\\gamma\\gys_{\\delta\\ssupface\\gamma}(x). \\]\n\nLet $a,b,s$ be three integers with $a\\equiv s\\pmod2$. Multiplying the previous equations by $\\sign(\\gamma,\\delta)$ and summing over couples $\\gamma\\ssubface\\delta$, we get that the three following diagrams are commutative.\n\\[ \\begin{tikzcd}[column sep=small]\n\\ST^{a,b,s}_1 \\rar[\"\\ell\"]\\dar[\"\\id\"] & \\ST^{a,b+2,s}_1 \\dar[\"\\id\"] \\\\\n\\ST^{a+2,b-2,s}_1 \\rar[\"\\ell\"] & \\ST^{a+2,b,s}_1\n\\end{tikzcd} \\quad \\begin{tikzcd}[column sep=small]\n\\ST^{a,b,s}_1 \\rar[\"\\ell\"]\\dar[\"\\i^*\"] & \\ST^{a,b+2,s}_1 \\dar[\"\\i^*\"] \\\\\n\\ST^{a+1,b,s+1}_1 \\rar[\"\\ell\"] & \\ST^{a+1,b+2,s+1}_1\n\\end{tikzcd} \\quad \\begin{tikzcd}[column sep=small]\n\\ST^{a,b,s}_1 \\rar[\"\\ell\"]\\dar[\"\\gys\"] & \\ST^{a,b+2,s}_1 \\dar[\"\\gys\"] \\\\\n\\ST^{a+1,b,s-1}_1 \\rar[\"\\ell\"] & \\ST^{a+1,b+2,s-1}_1\n\\end{tikzcd}\n\\]\nTo get the commutativity in $\\ST_1^{\\bul,\\bul}$, it remains to check that, in each diagram, if the top-left and bottom-right pieces appear in $\\ST_1^{\\bul,\\bul}$, then the two other pieces appear as well, which is immediate.\n\n\\medskip\n\nFor the second point, we have\n\\begin{gather*}\n\\ST_1^{a,b}\n  = \\bigoplus_{s\\geq\\abs a \\\\ s\\equiv a\\pmod2}\\bigoplus_{\\delta\\in X_\\f \\\\ \\dims\\delta=s} H^{a+b-s}(\\delta) \\\\\n\\ST_1^{a,2d-2a-b}\n  =\\bigoplus_{s\\geq\\abs a \\\\ s\\equiv a\\pmod2}\\bigoplus_{\\delta\\in X_\\f \\\\ \\dims\\delta=s} H^{2d-a-b-s}(\\delta)\n\\end{gather*}\nand $\\ell^{d-a-b}\\colon \\ST_1^{a,b}\\to\\ST_1^{a,{2d-2a-b}}$ can be rewritten as\n\\[ \\bigoplus_{s\\geq\\abs a \\\\ s\\equiv a\\pmod2}\\bigoplus_{\\delta\\in X_\\f \\\\ \\dims\\delta=s} (\\ell^\\delta)^{d-a-b}\\colon H^{a+b-s}(\\delta)\\longrightarrow H^{2d-a-b-s}(\\delta). \\]\nFor each $\\delta$ of dimension $s$, since $\\ell^\\delta$ is ample, we get an isomorphism\n\\[ (\\ell^\\delta)^{d-a-b}\\colon H^{a+b-s}(\\delta)\\longrightarrow H^{2d-a-b-s}(\\delta)=H^{2(d-\\dims\\delta)-(a+b-s)}.\\]\nThus, we conclude that $\\ell^{d-a-b}\\colon \\ST_1^{a,b}\\to\\ST_1^{a,{2d-2a-b}}$ is an isomorphism.\n\\end{proof}\n\n\n\n\\subsection{The Hodge-Lefschetz-primitive part $P^{a,b}$}\n\nFor $a \\leq 0$ and $b\\leq d-a$, define the \\emph{Hodge-Lefschetz} (or simply \\emph{HL}) \\emph{primitive part} $P^{a,b}$ of bidegree $(a,b)$ as\n\\[P^{a,b} \\colon = \\ST_1^{a, b} \\cap \\ker(N^{-a+1}) \\cap \\ker(\\ell^{d-a-b+1}).\\]\n\n\\begin{prop}\nWe have\n\\[P^{a,b} = \\bigoplus_{\\delta \\in X_\\f\\\\ \\dims\\delta = -a} P^{2a+b}(\\delta).\\]\n\\end{prop}\n\\begin{proof} For $b$ odd, both parts are zero. So suppose $b$ is even. In this case, we have\n\\[\\ST_1^{a, b} \\cap \\ker(N^{-a+1}) = \\bigoplus_{\\delta \\in X_\\f\\\\ \\dims\\delta = -a} H^{2a+b}(\\delta),\\]\nand thus\n\\[P^{a,b} = \\ker \\Bigl( \\bigoplus_{\\delta \\in X_\\f\\\\\n\\dims\\delta=-a} (\\ell^\\delta)^{d-a-b+1} \\colon   \\bigoplus_{\\delta \\in X_\\f\\\\ \\dims\\delta = -a} H^{2a+b}(\\delta) \\longrightarrow  \\bigoplus_{\\delta \\in X_\\f\\\\ \\dims\\delta = -a} H^{2d-b+2}(\\delta)\\Bigr).\\]\n\nTo conclude, note that since $\\dims\\delta =-a$, we have $d-a-b+1 = d-\\dims\\delta -(2a+b)+1$, and thus\n\\[\\ker\\Bigl(\\,(\\ell^\\delta)^{d-a-b+1} \\colon H^{2a+b}(\\delta) \\rightarrow H^{2d-b+2}(\\delta)\\Bigr) = P^{2a+b} (\\delta),\\]\nand the proposition follows.\n\\end{proof}\n\n\n\n\\subsection{Hodge diamond}\n\n\\begin{figure}\n\\caption{The tropical Hodge diamond} \\label{fig:hodge_diamond}\n\\hodgediamond\n\\end{figure}\n\nBy what we proved in the previous sections, $\\ST_1^{\\bul,\\bul}$ forms a \\emph{Hodge diamond} as illustrated in Figure \\ref{fig:hodge_diamond}. Let us explain what this means.\n\n\\medskip\n\nWe do a change of coordinates $(a,b) \\to (a, d-a-b)$ to form\n\\[\\HD^{a,d-a-b}:=\\ST_1^{a,b}.\\]\nThe result of this transformation is that the nonzero terms of $\\HD^{\\bul,\\bul}$ now form a rhombus.\n\nThe horizontal symmetry maps a piece of $\\HD^{\\bul,\\bul}$ onto an isomorphic piece by applying the monodromy operator $N$ enough number of times. More precisely, for an integer $a\\geq0$, we get\n\\[ N^a\\colon \\HD^{-a,b} \\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} \\HD^{a,b} \\]\nSimilarly, the vertical symmetry maps a piece of $\\HD^{\\bul,\\bul}$ onto another isomorphic piece by applying the Lefschetz operator $\\ell$ enough number of times: for $b\\geq0$,\n\\[ \\ell^b\\colon \\HD^{a,-b} \\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} \\HD^{a,b}. \\]\nIn the figure, the four squares in both diagrams, in $\\ST_1^{\\bul,\\bul}$ and $\\HD^{\\bul,\\bul}$, represent four isomorphic pieces. Moreover, the central symmetry corresponds to the Poincar\u00e9 duality $\\HD^{-a,-b}\\simeq H^{a,b}$.\n\nTo justify the name of tropical Hodge diamond, on the bottom of the figure is shown the tropical Hodge groups to which the Steenbrink diamond degenerates. We show in Theorem \\ref{thm:main} that the symmetries described above also hold for the tropical Hodge decomposition, as in the usual Hodge diamond.\n\n\\medskip\n\nFinally, notice that $\\HD_{\\bul,\\bul}$ defined by $\\HD_{a,b}=\\HD^{-a,-b}$ is a HL-structure as defined in Section \\ref{sec:differential_HL_structure}, \\cf. Definition \\ref{defi:HL-structure}.\n\n\n\n\\subsection{Polarization on $\\ST_1^{\\bul,\\bul}$} We now describe a natural polarization on $\\ST_1^{\\bul, \\bul}$. We prove in the next section that it induces a positive definite symmetric bilinear form on the HL-primitive parts $P^{a,b}$ defined above.\n\nWe first define a bilinear form $\\psi$ on $\\ST_1^{\\bul, \\bul}$ as follows. Let $x \\in \\ST_1^{a,b,s}$ and $y \\in \\ST_1^{a',b',s'}$. Write\n\\[ x=\\sum_{\\delta\\in X_\\f \\\\ \\dims\\delta=s} x_\\delta \\]\nwith $x_\\delta\\in H^{a+b-s}(\\delta)$, and similarly for $y$. We define the \\emph{degree map}\n\\[ \\deg(x\\cdot y) := \\begin{cases}\n\\sum_{\\delta\\in X_\\f \\\\ \\dims\\delta=s}\\deg(x_\\delta\\cdot y_\\delta) & \\text{if $s=s'$}, \\\\\n0 & \\text{otherwise}.\n\\end{cases} \\]\nThe bilinear form $\\psi$ is then defined by\n\\begin{equation}\n\\psi(x,y) := \\begin{cases}\n\\epsilon(a,b) \\deg(x \\cdot y)  & \\textrm{if $a+a'=0$, \\ $b+b'=2d$, and $s'=s$,} \\\\\n0 & \\textrm{otherwise},\n\\end{cases}\n\\end{equation}\nwhere for a pair of integers $a,b$, with $b$ even,\n\\[\\epsilon(a,b) := (-1)^{ a+ \\frac b2},\\]\nand otherwise, if $b$ is odd, $\\epsilon(a,b) =1$ (but in this case, we necessarily have $x=0$).\n\n\\medskip\n\nThe following proposition gathers basic properties of the the bilinear pairing $\\psi$.\n\n\\begin{prop} \\label{prop:properties_psi}\nWe have for all pairs of elements $x, y \\in \\bigoplus_{a,b} \\ST_1^{a,b}$,\n\\begin{enumerate}[label={\\textnormal{(\\textit{\\roman*})}}]\n\\item \\label{enum:psi:i} $\\psi(x,y) = (-1)^d \\psi(y,x)$.\n\\item \\label{enum:psi:ii} $\\psi(Nx, y) + \\psi(x, Ny) =0$.\n\\item \\label{enum:psi:iii} $\\psi(\\ell x, y) + \\psi(x, \\ell y) =0$.\n\\item \\label{enum:psi:iv} $\\psi(\\i^*x, y) + \\psi(x, \\gys y) =0$.\n\\item \\label{enum:psi:v} $\\psi(\\gys x, y) + \\psi(x, \\i^* y) =0$.\n\\item \\label{enum:psi:vi} $\\psi(\\d x, y) + \\psi(x, \\d y) =0$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nLet $x\\in\\ST^{a,b,s}_1$ and $y\\in\\ST_1^{a',b',s'}$ be two nonzero elements for integers $a,b,s, a',b',s'$ with $s\\geq\\abs{a}$ and $s'\\geq\\abs{a'}$.\nWe proceed by verifying the statements point by point.\n\n\\medskip\n\n\\begin{enumerate}[label={\\textnormal{(\\textit{\\roman*})}}, leftmargin=0pt]\n\\item if $a+a'\\neq 0$ or $b+b'\\neq 2d$, then both $\\psi(x,y)$ and $\\psi(y,x)$ are zero. Otherwise,\n\\[ \\epsilon(a',b')=\\epsilon(-a,2d-b)=(-1)^{-a+\\frac{2d-b}2}=(-1)^d\\epsilon(a,b). \\]\nThus, $\\psi(x,y) = (-1)^d \\psi(y,x)$.\n\n\\medskip\n\n\\item Recall that $N$ is a map of bidegree $(2,-2)$. If $a+a'+2\\neq 0$ or $b+b'-2\\neq 2d$ or $s\\neq s'$, then both $\\psi(Nx,y)$ and $\\psi(x,Ny)$ are zero. Otherwise, $s=s'\\geq\\abs{a'}=\\abs{a+2}$. Thus $Nx\\neq 0$. By symmetry, $Ny\\neq 0$. Since $N$ acts as identity on both $x$ and $y$, we get $\\deg(Nx\\cdot y)=\\deg(x\\cdot Ny)$. Moreover, $\\epsilon(a+2,b-2)=-\\epsilon(a,b)$. Hence, we obtain $\\psi(Nx, y) + \\psi(x, Ny) =0$.\n\n\\medskip\n\n\\item Recall that $\\ell$ is a map of bidegree $(0,2)$. If $a+a'\\neq 0$ or $b+b'+2\\neq 2d$ or $s\\neq s'$, then both $\\psi(\\ell x,y)$ and $\\psi(x,\\ell y)$ are zero. Otherwise,\n\\[ \\deg(\\ell x\\cdot y)=\\sum_{\\delta\\in X_\\f \\\\ \\dims\\delta = s}\\deg(\\ell^\\delta\\cdot x_\\delta\\cdot y_\\delta)=\\deg(x\\cdot \\ell y). \\]\nMoreover, $\\epsilon(a,b+2)=-\\epsilon(a,b)$. Thus, we get $\\psi(\\ell x, y) + \\psi(x,\\ell y)=0$.\n\n\\medskip\n\n\\item If $a+a'+1\\neq0$ or $b+b'\\neq2d$ or $s+1\\neq s'$, then $\\psi(\\i^*x,y)$ and $\\psi(x,\\gys y)$ are both zero. Otherwise, $s'-1=s\\geq\\abs a=\\abs{a'+1}$. Thus, the Gysin map $\\gys\\colon\\ST_1^{a',b',s'}\\to\\ST_1^{a'+1,b',s'-1}$ has a codomain belonging to $\\ST_1^{\\bul,\\bul}$. This also holds for $\\i^*$. Moreover, using Equation \\eqref{eqn:i_gys_dual}, we get\n\\begin{align*}\n\\deg(\\i^*(x)\\cdot y)\n  &= \\sum_{\\delta\\in X_\\f \\\\ \\dims\\delta=s+1}\\sum_{\\gamma\\in X_\\f \\\\ \\gamma\\ssubface\\delta}\\deg(\\sign(\\gamma,\\delta)\\i^*_{\\gamma\\ssubface\\delta}(x_\\gamma)\\cdot y_\\delta) \\\\\n  &= \\sum_{\\gamma\\in X_\\f \\\\ \\dims\\gamma=s}\\sum_{\\delta\\in X_\\f \\\\ \\delta\\ssupface\\gamma}\\deg(x_\\gamma\\cdot\\sign(\\gamma,\\delta)\\gys_{\\delta\\ssupface\\gamma}(y_\\delta)) \\\\\n  &=\\deg(x\\cdot\\gys(y)).\n\\end{align*}\nSince $\\epsilon(a+1,b)=-\\epsilon(a,b)$, we get $\\psi(\\i^*x,y)+\\psi(x,\\gys y)=0$.\n\n\\medskip\n\n\\item The equality $\\psi(\\gys x, y) + \\psi(x, \\i^* y) = 0$ follows from the points \\ref{enum:psi:i} and \\ref{enum:psi:iv}.\n\n\\medskip\n\n\\item Summing up the equalities in \\ref{enum:psi:iv} and \\ref{enum:psi:v} gives $\\psi(\\d x, y) + \\psi(x, \\d y) =0$ by the definition of the differential $\\d$. \\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\n\n\\subsection{Induced polarization on the Hodge-Lefschetz primitive parts}\n\nThe bilinear form $\\psi$ induces a positive definite symmetric bilinear form on the Hodge-Lefschetz primitive parts $P^{a,b}$ that we now describe.\n\n\\begin{prop} \\label{prop:primitive_polarization}\nFor integer $a\\leq 0$ and $b\\leq d-a$, the bilinear form $\\psi(\\,\\cdot\\,, \\ell^{d-a-b}N^{-a}\\,\\cdot\\,)$ is symmetric positive definite on $P^{a,b}$. More precisely, it is the orthogonal sum of polarizations $(-1)^{a+b\/2}Q_{\\delta} (\\,\\cdot\\,,\\cdot\\,)$ for $\\delta \\in X_\\f$ on the primitive part $P^{2a+b}(\\delta)$, under the identification\n\\[P^{a,b} = \\bigoplus_{\\delta \\in X_\\f \\\\ \\dims\\delta =-a} P^{2a+b} (\\delta).\\]\n\\end{prop}\n\n\\begin{proof}\nBy the properties of the monodromy operator, the isomorphism\n\\[ N^{-a}\\colon \\ST_1^{a,2d-2a-b} \\to \\ST_1^{-a, 2d-b} \\]\nis the identity map on each isomorphic piece $H^{2a+b}(\\delta)$ of $\\ST_1^{a,2d-2a-b}$ and $\\ST_1^{-a, 2d-b}$, for any $\\delta \\in X_\\f$ with $\\dims\\delta =-a$.\n\nBy definition of $\\psi$, for a pair of elements $x, y \\in P^{2a+b}(\\delta)$, we get\n\\begin{align*}\n\\psi(x,\\ell^{d-a-b}N^{-a}y)\n  &= \\psi(x,N^{-a}\\ell^{d-a-b}y) \\\\\n  &= \\epsilon(a,b) \\deg(x\\cdot\\ell^{d-a-b}y) \\\\\n  &= (-1)^{a+b\/2}\\deg(x\\cdot\\ell^{d-a-b}y) \\\\\n  &= (-1)^{a+b\/2} Q_\\delta(x,y).\n\\end{align*}\nThis concludes the proof since $(-1)^{a+b\/2}Q_\\delta$ is symmetric positive definite by local Hodge-Riemann relations.\n\\end{proof}\n\n\n\n\\subsection{Hodge-Lefschetz decomposition}\n\nFrom the preceding discussion, we get the following theorem. It can be illustrated as in Figure \\ref{fig:hodge_diamond}: the black square can be decomposed as the direct sum of primitive parts lying in the grey area.\n\n\\begin{thm} \\label{thm:ST_1_decomposition}\nFor each pair of integers $a\\leq0$ and $b\\leq d-a$, we have the decomposition\n\\[ \\ST_1^{a,b} = \\bigoplus_{s,r \\geq 0} \\ell^r N^s P^{a-2s, b+2s-2r}. \\]\n\\end{thm}\n\n\\begin{proof}\nWe give here a direct proof based on the definition of $\\ST_1^{\\bul,\\bul}$ and using local Lefschetz decompositions. We however note that this theorem can be proved thanks to the general theory of HL-structures, \\cf. Proposition \\ref{prop:HL_decomposition}. We have\n\\begin{align*}\n\\ST_1^{a,b}\n  &= \\bigoplus_{s \\geq 0} \\ST_1^{a,b,2s-a} \\\\\n  &= \\bigoplus_{s \\geq 0} N^s \\ST_1^{a-2s,b+2s,2s-a} \\\\\n  &= \\bigoplus_{s \\geq 0} N^s \\bigoplus_{\\delta \\in X_\\f \\\\ \\dims\\delta = 2s-a} \\underbrace{H^{2a+b-2s}(\\delta)}_{\\subseteq \\ST_1^{a-2s, b+2s}} \\\\\n  &= \\bigoplus_{s \\geq 0} N^s \\bigoplus_{\\delta \\in X_\\f \\\\ \\dims\\delta = 2s-a} \\bigoplus_{r \\geq 0} \\ell^r \\underbrace{P^{2a+b-2s-2r}(\\delta)}_{\\subseteq\\ST_1^{a-2s, b+2s-2r}} \\\\\n  &= \\bigoplus_{r,s \\geq 0} \\ell^r N^s \\bigoplus_{\\delta \\in X_\\f \\\\ \\dims\\delta = 2s-a} P^{2a+b-2s-2r}(\\delta), \\\\\n\\ST_1^{a,b}\n  &= \\bigoplus_{r,s \\geq 0} \\ell^r N^s P^{a-2s, b+2s-2r}. \\qedhere\n\\end{align*}\n\\end{proof}\n\n\n\n\\subsection{Polarized Hodge-Lefschetz structure on the cohomology of $(\\ST_1^{\\bul, b}, \\d)$}\n\nIn this section we show that when passing to the cohomology, the operators $N$, $\\ell$ and the polarization $\\psi$ induce a polarized Hodge-Lefschetz structure on the cohomology groups $L^{a,b} := H^{a}(\\ST_1^{\\bul, b}, \\d)$. By this we mean the following.\n\n\\medskip\n\nFirst, since $[\\d, \\ell] =0$ and $[\\d,N]=0$, we get induced maps\n\\[\\ell \\colon L^{a,b} \\to L^{a, b+2}\\]\nand\n\\[N\\colon L^{a,b} \\to L^{a+2, b-2}\\]\nsuch that $[N,\\ell] = 0$.\n\nSecond, since $\\psi(\\d\\,\\cdot\\,,\\cdot\\,)=-\\psi(\\,\\cdot\\,,\\d\\,\\cdot\\,)$, we get an induced pairing\n\\[ \\psi\\colon L^{\\bul,\\bul}\\times L^{\\bul,\\bul} \\to \\corps. \\]\n\nMoreover, we will show that\n\\begin{thm} \\label{thm:tropical_differential_HL_structure}We have the following properties.\n\\begin{itemize}\n\\item For $a\\leq 0$, the map $N^{-a} \\colon L^{a,b} \\to L^{-a,b}$ is an isomorphism.\n\\item For $a,b$ with $d-a-b \\geq 0$, the map $\\ell^{d-a-b} \\colon L^{a,b} \\to L^{a, 2d-2a-b}$ is an isomorphism.\n\\item Let $a\\leq 0$ and $b \\leq d-a$ and denote by $L_0^{a,b}$ the HL-primitive part of $L^{a,b}$ defined by\n\\[ L_0^{a,b} = L^{a,b} \\cap \\ker(N^{-a+1}) \\cap \\ker(\\ell^{d-a-b+1}).\\]\nThe polarization $\\psi$ induces a symmetric bilinear form $\\psi(\\,\\cdot\\,,\\ell^{d-a-b}N^{-a} \\,\\cdot\\,)$ which is positive definite on $L_0^{a,b}$.\n\\item For integers $a\\leq 0$ and $b\\leq d-a$, we have the decomposition\n\\[ L^{a,b} = \\bigoplus_{s,r \\geq 0} \\ell^r N^s L^{a-2s, b+2s-2r}_0.\\]\nMoreover, this decomposition is orthogonal for the pairing $\\psi(\\,\\cdot\\,,\\ell^{d-a-b}N^{-a}\\,\\cdot\\,)$.\n\\end{itemize}\n\\end{thm}\n\n\\begin{proof}\nWe can use the change of coordinates $\\HD_{-a,-(d-a-b)}:=\\ST_1^{a,b}$ for any integers $a,b$. By the previous discussion, we get that $(\\HD_{-a,-(d-a-b)},N,\\ell,\\phi,\\d)$ is a differential polarized HL-structure in the sense of Section~\\ref{sec:differential_HL_structure}. We can therefore apply Theorem \\ref{thm:differential_HL-structure} to conclude.\n\\end{proof}\n\n\n\n\\subsection{Singular monodromy operator}\n\nWe define the \\emph{singular tropical monodromy operator} as follows. For each face $\\delta\\in X_\\f$, let $o_\\delta$ be a point of $\\TT\\delta\\subseteq N_\\mathbb{R}$. For instance, we can take $o_\\delta$ to be the centroid of $\\delta$. Let $\\gamma$ be a face in $X_\\f$ of dimension $q$. Then we set\n\\[ \\begin{array}{rccc}\nN\\colon & C^{p,q}(X) & \\to & C^{p-1,q+1}(X), \\\\\n   & \\alpha\\in F^p(\\gamma) & \\mapsto & \\underset{\\substack{\\delta\\in X_\\f\\\\\n   \\delta \\ssupface \\gamma}}{\\bigoplus} \\,\\,\\alpha\\bigr(\\,\\cdot\\, \\wedge (o_\\delta-o_\\gamma)\\bigr).\n\\end{array}\n\\]\n\n\\begin{thm} \\label{thm:tropical_monodromy}\nThe singular tropical monodromy operator corresponds to the monodromy operator on $\\ST_1^{\\bul,\\bul}$ via the isomorphism of Theorem \\ref{thm:steenbrink}.\n\\end{thm}\n\n\\begin{proof}\nThis is a direct consequence of Proposition \\ref{prop:com:N_commutes}.\n\\end{proof}\n\n\\begin{cor} The operator $N\\colon H^{p,q} \\to H^{p-1, q+1}$ coincides with the eigenwave operator $\\phi$ constructed in~\\cite{MZ}.\n\\end{cor}\n\\begin{proof} One can show that the singular tropical monodromy operator coincides with the eigenwave operator. So the statement follows from Theorem~\\ref{thm:tropical_monodromy}.\n\\end{proof}\n\n\\begin{remark} It follows that the monodromy operator $N$ coincides as well with the monodromy operator defined on the level of Dolbeault cohomology groups~\\cites{Liu19}. This is a consequence of \\cite{Jell19} which relates the eigenwave operator to the monodromy on Dolbeault cohomology.\n\\end{remark}\n\n\n\n\\subsection{HL, HR and Monodromy for tropical cohomology}\n\nFrom the result in the previous section, we get the following theorem.\n\n\\begin{thm}\\label{thm:main} We have\n\\begin{itemize}\n\\item \\emph{(Weight-monodromy conjecture)} For $q>p$ two non-negative integers, we get an isomorphism\n\\[N^{q-p} \\colon H_{\\trop}^{q,p}(X) \\to H_{\\trop}^{p,q}(X).\\]\n\n\\item \\emph{(Hard Lefschetz)} For $p+q \\leq d\/2$, the Lefschetz operator $\\ell$ induces an isomorphism\n\\[\\ell^{d- p-q}\\colon H_{\\trop}^{p,q}(X) \\to H^{d-q, d-p}_{\\trop}(X).\\]\n\n\\item \\emph{(Hodge-Riemann)} The pairing $(-1)^p \\bigl< \\,\\cdot\\,,\\, \\ell^{d-p-q} N^{q-p} \\,\\cdot\\, \\bigr>$ induces a positive-definite pairing on the primitive part $P^{q,p}$, where $\\bigl< \\,\\cdot\\,,\\cdot\\, \\bigr>$ is the natural pairing\n\n\\[\\bigl< \\,\\cdot\\,, \\cdot\\,\\bigr> \\colon H^{q,p}(X) \\otimes H^{d-q,d-p}(X) \\to H^{d,d}(X) \\simeq \\mathbb Q.\\]\n\\end{itemize}\n\\end{thm}\n\n\n\n\\subsection{Hodge index theorem for tropical surfaces}\nIn this final part of this section, we explain how to deduce the Hodge index theorem for tropical surfaces from Theorem~\\ref{thm:main}.\n\\begin{proof}[Proof of Theorem~\\ref{thm:hodgeindex}] The primitive part decomposition theorem implies that we can decompose the cohomology group $H^{1,1}_\\trop(X, \\mathbb{Q})$ into the direct sum\n\\[H^{1,1}_\\trop(X, \\mathbb{Q}) = \\ell\\, H^{0,0}_\\trop(X, \\mathbb{Q}) \\oplus N\\, H^{2,0}_\\trop(X, \\mathbb{Q}) \\oplus H^{1,1}_\\prim(X, \\mathbb{Q}),\\]\nwhere $H^{1,1}_\\prim(X, \\mathbb{Q}) = \\ker(\\ell) \\cap \\ker(N)$. Moreover, by Hodge-Riemann, the pairing is positive definite on\n$\\ell\\, H^{0,0}_\\trop(X, \\mathbb{Q})$ and $N\\, H^{2,0}_\\trop(X, \\mathbb{Q})$, and it is negative definite on $H^{1,1}_\\prim(X, \\mathbb{Q})$.\n\nBy Poinca\\'re duality, we have $H^{2,0}_\\trop(X, \\mathbb{Q}) \\simeq H^{0,2}_\\trop(X, \\mathbb{Q})$. By definition of tropical cohomology, since $F^0$ is the constant sheaf $\\mathbb{Q}$, we have $H^{0,2}_\\trop(X, \\mathbb{Q}) \\simeq H^2(\\mathfrak X,\\mathbb{Q})$. Moreover, we have $H^{0,0}_\\trop(X, \\mathbb{Q}) \\sim \\mathbb{Q}$.\nWe conclude that the signature of the intersection pairing is given by $(1+ b_2, h^{1,1}- 1-b_2)$, as stated by the Hodge index theorem.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Hodge-Lefschetz structures}\n\\label{sec:differential_HL_structure}\n\nIn this section, we prove that the homology of a differential Hodge-Lefschetz structure is a polarized Hodge-Lefschetz structure, thus finishing the proof of Theorem~\\ref{thm:tropical_differential_HL_structure} and the main theorem of the previous section.\n\n\\medskip\n\nThe main references for differential Hodge-Lefschetz structures are Saito's work on Hodge modules~\\cite{Saito}, the paper by Guill\\'en and Navarro Aznar on invariant cycle theorem~\\cite{GNA90} and the upcoming book by Sabbah and Schnell~\\cite{SabSch} on mixed Hodge modules, to which we refer for more information. We note however that our set-up is slightly different, in particular, our differential operator is skew-symmetric with respect to the polarization. The proof in~\\cite{SabSch} is presented also for the case of mono-graded complexes and is based on the use of representation theory of $\\mathrm{SL}_2(\\mathbb{R})$ as in~\\cite{GNA90}. So we give a complete self-contained proof of the main theorem. Our proof is direct and does not make any recourse to representation theory, although it might be possible to recast in the language of representation theory the final combinatorial calculations we need to elaborate.\n\n\\medskip\n\nIn order to simplify the computations, it turns out that it will be more convenient to work with coordinates different from the ones in the previous section, and that is what we will be doing here. The new coordinates are compatible with the ones in~\\cite{SabSch}.\n\n\n\n\\subsection{Hodge-Lefschetz structure}\n\n\\begin{defi} \\label{defi:HL-structure}\nA \\emph{(bigraded) Hodge-Lefschetz structure}, or more simply an \\emph{HL-structure}, $(H_{\\bul,\\bul}, N_1, N_2)$ is a bigraded finite dimensional vector space $H_{\\bul,\\bul}$ endowed with two endomorphisms $N_1$ and $N_2$ of respective bidegree $(-2,0)$ and $(0,-2)$ such that,\n\\begin{itemize}\n\\item $[N_1,N_2]=0$,\n\\item for any pair of integers $a,b$ with $a\\geq 0$, we have $N_1^a\\colon H_{a,b}\\to H_{-a,b}$ is an isomorphism,\n\\item for any pair of integers $a, b$ with $b\\geq0$, we have $N_2^b\\colon H_{a,b}\\to H_{a,-b}$ is an isomorphism.\n\\end{itemize}\n\\end{defi}\n\nFor the rest of this section, we fix a HL-structure $(H_{\\bul,\\bul}, N_1, N_2)$.\n\n\\medskip\n\nWe define the \\emph{primitive parts} of $H_{\\bul,\\bul}$ by\n\\[\\forall \\, a,b\\geq 0,\\qquad P_{a,b}:=H_{a,b}\\cap\\ker(N_1^{a+1})\\cap\\ker(N_2^{b+1}). \\]\n\nWe have the following decomposition into primitive parts.\n\n\\begin{prop} \\label{prop:HL_decomposition}\nFor each pair of integers $a,b\\geq0$, we have the decomposition\n\\[ H_{a,b} = \\bigoplus_{s,r \\geq 0} N_1^r N_2^s P_{a+2r, b+2s}. \\]\n\\end{prop}\n\nDenote by $H^+_{\\bul,\\bul}$ the bigraded subspace\n\\[ H^+_{\\bul,\\bul}=\\bigoplus_{a,b\\geq0}H_{a,b}. \\]\nDefine $N_1^\\dual\\colon H^+_{\\bul,\\bul}\\to H^+_{\\bul,\\bul}$ of bidegree $(2,0)$, by\n\n\\[ N_1^\\dual\\rest{H_{a,b}}=\\Bigl(\\bigl(N_1\\rest{H_{a+2,b}}\\bigr)^{a+2}\\Bigr)^{-1}\\Bigl(N_1\\rest{H_{a,b}}\\Bigr)^{a+1}\\qquad\\forall a,b\\geq0. \\]\n\n\\medskip\n\nIt verifies the following properties. Let $a,b\\geq0$. Then,\n\n\\[ N_1^\\dual N_1\\rest{H_{a+2,b}}=\\id\\rest{H_{a+2,b}} \\quad\\text{ and }\\quad \\ker(N_1^\\dual)\\cap H_{a,b}=\\ker(N_1^{a+1})\\cap H_{a,b}. \\]\n\nIn particular, $N_1\\colon H_{a+2,b}\\to H_{a,b}$ is injective, and $N_1^\\dual\\colon H_{a,b}\\to H_{a+2,b}$ is surjective.\n\nFrom this, we can deduce that the map $N_1N_1^\\dual\\rest{H_{a,b}}$ is the projection along $\\ker(N_1^{a+1})\\cap H_{a,b}$ onto $N_1H_{a+2,b}$, i.e., for the decomposition\n\\[H_{a,b} =  N_1H_{a+2,b} \\oplus \\Bigl(\\ker(N_1^{a+1})\\cap H_{a,b}\\Bigr).\\]\n\n\\medskip\n\nWe define $N_2^\\dual$ in a similar way, and note that it verifies similar properties. In particular, $N_2N_2^\\dual\\rest{H_{a,b}}$ is the projection along $\\ker(N_2^{b+1})\\cap H_{a,b}$ onto $N_2H_{a,b+2}$ for the decomposition\n\\[H_{a,b} =  N_2H_{a,b+2} \\oplus \\Bigl(\\ker(N_2^{b+1})\\cap H_{a,b}\\Bigr).\\]\n\n\\medskip\n\nMoreover, from the commutativity of $N_1$ and $N_2$, more precisely, from\n\\[ [N_1^{a+2},N_2]\\rest{H_{a+2,b+2}}=[N_1,N_2^{b+2}]\\rest{H_{a+2,b+2}}=[N_1^{a+2},N_2^{b+2}]\\rest{H_{a+2,b+2}}=0, \\]\nwe infer that\n\\[ [N_1^\\dual,N_2]\\rest{H_{a,b+2}}=[N_1,N_2^\\dual]\\rest{H_{a+2,b}}=[N_1^\\dual,N_2^\\dual]\\rest{H_{a,b}}=0. \\]\nIn particular,\n\\[ N_1N_1^\\dual N_2N_2^\\dual=N_2N_2^\\dual N_1N_1^\\dual=N_1N_2N_1^\\dual N_2^\\dual. \\]\nNotice that $N_1N_2N_1^\\dual N_2^\\dual\\rest{H_{a,b}}$ is a projection onto $N_1N_2H_{a+2,b+2}$.\n\n\\medskip\n\nWe come back to the proof of the proposition.\n\\begin{proof}[Proof of Proposition~\\ref{prop:HL_decomposition}]\nClearly, $N_1\\colon H_{a+2,b}\\to H_{a,b}$ is injective for $a\\geq0$. Similarly for $N_2$. Therefore, proceeding by induction, it suffices to prove that for any pair of integers $a,b\\geq0$, we have\n\\begin{gather*}\nH_{a,b}= \\bigl(N_1H_{a+2,b} + N_2H_{a,b+2}\\bigr) \\oplus P_{a,b} \\quad  \\text{ and } \\quad N_1H_{a+2,b} \\cap N_2H_{a,b+2} = N_1N_2H_{a+2,b+2}.\n\\end{gather*}\nThis can be deduced from the following lemma applied to $N_1N_1^\\dual\\rest{H_{a,b}}$ and $N_2N_2^\\dual\\rest{H_{a,b}}$ and the corresponding decompositions of $H_{a,b}$.\n\\end{proof}\n\n\\begin{lemma}\nLet $V$ be a finite dimensional vector space. For $i\\in\\{1,2\\}$, suppose we have decompositions $V = I_1 \\oplus K_1 = I_2 \\oplus K_2$ for subspaces $I_1, K_1, I_2, K_2 \\subseteq V$. Denote by $\\pi_i\\colon V\\to V$ the projection of $V$ onto $I_i$ along $K_i$. Then, if $\\pi_1$ and $\\pi_2$ commute, we get\n\\[ V=I_1\\cap I_2 \\woplus I_1\\cap K_2 \\woplus K_1\\cap I_2 \\woplus K_1\\cap K_2\\quad \\text{ and } \\quad I_1\\cap I_2=\\Im(\\pi_1\\circ\\pi_2).\\]\n\\end{lemma}\n\n\\begin{proof}\nSince $\\pi_1$ and $\\pi_2$ commute, $\\pi_2(I_1)\\subseteq I_1$. Thus, $\\pi_2$ can be restricted to $I_1$. This restriction is a projection onto $I_2 \\cap I_1$ along $K_2 \\cap I_1$. Thus,\n\\[ I_1=I_2 \\cap I_1 \\woplus K_2 \\cap I_1 \\quad\\text{ and }\\quad I_1\\cap I_2=\\Im(\\pi_2\\circ\\pi_1). \\]\nApplying the same argument to the commuting projections $\\id-\\pi_1$ and $\\id-\\pi_2$, we get that\n\\[ K_1=K_2 \\cap K_1 \\woplus I_2 \\cap K_1. \\]\nHence,\n\\[ V = I_1 \\oplus K_1 = I_1\\cap I_2 \\woplus I_1\\cap K_2 \\woplus K_1\\cap I_2 \\woplus K_1\\cap K_2. \\qedhere \\]\n\\end{proof}\n\n\n\n\\subsection{Polarization $\\psi$ and its corresponding scalar product $\\phi$}\n\nA \\emph{polarization} on the HL-structure $H_{\\bul,\\bul}$ is a bilinear form $\\psi(\\,\\cdot\\,,\\cdot\\,)$ such that:\n\\begin{itemize}\n\\item for any integers $a,b,a',b'$ and any elements $x\\in H_{a,b}$ and $y\\in H_{a',b'}$, the pairing $\\psi(x,y)$ is zero unless $a+a'=b+b'=0$,\n\\item for any $a,b\\geq0$, the pairing $\\psi(\\,\\cdot\\,,N_1^aN_2^b\\,\\cdot\\,)$ is symmetric definite positive on $P_{a,b}$,\n\\item $N_1$ and $N_2$ are skew-symmetric for $\\psi$.\n\\end{itemize}\n\nFor the rest of this section, we assume that $H_{\\bul,\\bul}$ is endowed with a polarization $\\psi$.\n\n\\medskip\n\nDenote as in the previous section $H^+_{\\bul,\\bul}$ the bigraded subspace\n\\[ H^{+}_{\\bul,\\bul}=\\bigoplus_{a,b\\geq0}H_{a,b}. \\]\n\nWe define the operator $\\w$ on $H_{\\bul,\\bul}$ as follows.\n\nLet $a,b\\geq0$ and $r,s\\geq0$ be four integers. Let $x\\in P_{a,b}$. Then, we set\n\\[ \\w N_1^rN_2^sx:=\\begin{cases}\n(-1)^{r+s}\\frac{r!}{(a-r)!}\\frac{s!}{(b-s)!}N_1^{a-r}N_2^{b-s}x & \\text{if $r\\leq a$ and $s\\leq b$} \\\\\n0 & \\text{otherwise.}\n\\end{cases} \\]\n\nUsing the decomposition into sum of primitive parts from the previous section, one can verify that $\\w$ is well-defined.\n\nWe will provide some explanations about the choice of the operator $\\w$, in particular about the choice of the renormalization factors, at the beginning of Section \\ref{subsec:Laplacian_commute}.\n\n\\medskip\n\nSet $\\phi(\\,\\cdot\\,,\\cdot\\,):=\\psi(\\,\\cdot\\,,\\w\\,\\cdot\\,)$.\n\n\\begin{prop}\nWe have the following properties.\n\\begin{itemize}\n\\item The operator $\\w$ is invertible with its inverse verifying\n\\[ \\w^{-1}N_1^rN_2^sx=(-1)^{a+b}\\w N_1^rN_2^sx. \\]\n\\item The pairing $\\phi$ is symmetric positive definite.\n\\item The decomposition into primitive parts of $H_{\\bul,\\bul}$ is orthogonal for $\\phi$. More precisely, let $a,b,r,s\\geq0$ and $a',b',r',s'\\geq0$ be eight integers and let $x\\in P_{a,b}$ and $y\\in P_{a',b'}$. Then,\n\\[ \\phi(N_1^rN_2^sx, N_1^{r'}N_2^{s'}y)=0 \\text{ unless $r=r', s=s', a=a'$ and $b=b'$}. \\]\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nThe first point is an easy computation. For the two other points, take $x\\in P_{a,b}$ and $y\\in P_{a',b'}$. We can assume that $r\\leq a$, $s\\leq b$, $r'\\leq a'$ and $s'\\leq b'$. Set \\[ C=\\frac{r'!\\,s'!}{(a'-r')!\\,(b'-s')!}\\,. \\]\nWe have\n\\begin{align*}\n\\phi(N_1^rN_2^sx, N_1^{r'}N_2^{s'}y)\n  &= \\psi(N_1^rN_2^sx, \\w N_1^{r'}N_2^{s'}y) \\\\\n  &= (-1)^{r'+s'}C\\psi(N_1^rN_2^sx, N_1^{a'-r'}N_2^{b'-s'}y) \\\\\n  &= (-1)^{a'+b'}C\\psi(N_1^{a'-r'+r}N_2^{b'-s'+s}x, y) \\\\\n  &= (-1)^{r'+s'+r+s}C\\psi(x, N_1^{a'-r'+r}N_2^{b'-s'+s}y).\n\\end{align*}\nSuppose the value of the above pairing is nonzero. Then, since $x\\in H_{a,b}$ and\n\\[ N_1^{a'+r-r'}N_2^{b'+s-s'}y\\in H_{-a'+2r'-2r,-b'+2s'-2s}, \\]\nwe should have $a-a'+2r'-2r=0$. Moreover, since $x\\in\\ker(N_1^{a+1})$ and $y\\in\\ker(N_1^{a'+1})$, we get\n\\[ a'-r'+r\\leq a \\quad\\text{ and }\\quad a'-r'+r\\leq a'. \\]\nHence, we get\n\\[ r-r'\\leq a-a'=2(r-r') \\quad\\text{ and }\\quad r-r'\\leq 0,\\]\nwhich in turn implies that $r-r'=a-a'=0$, i.e., $r=r'$ and $a=a'$. By symmetry, we get $s=s'$ and $b=b'$. We infer that\n\\[ \\phi(N_1^rN_2^sx, N_1^rN_2^sy) = C\\psi(x, N_1^aN_2^by), \\]\nwhich concludes the proof since $\\psi(\\,\\cdot\\,,N_1^aN_2^b\\,\\cdot\\,)$ is symmetric positive definite on $P_{a,b}$.\n\\end{proof}\n\n\n\n\\subsection{Differential polarized HL-structure}\n\nA \\emph{differential} on the polarized HL-structure $H_{\\bul,\\bul}$ is an endomorphism $\\d$ of bidegree $(-1,-1)$ such that\n\\begin{itemize}\n\\item $\\d^2=0$,\n\\item $[\\d,N_1]=[\\d,N_2]=0$,\n\\item $\\d$ is skew-symmetric for $\\psi$.\n\\end{itemize}\n\nIn the rest of this section, we fix a differential $\\d$ on $H_{\\bul,\\bul}$.\n\n\\begin{prop}\nThe following holds for any pair of integers $a,b\\geq0$:\n\\[ \\d P_{a,b} \\subseteq P_{a-1,b-1} \\oplus N_1 P_{a+1,b-1} \\oplus N_2 P_{a-1,b+1} \\oplus N_1N_2 P_{a+1,b+1}, \\]\nwhere, by convention, $P_{a',b'}=0$ if $a'<0$ or $b'<0$.\n\\end{prop}\n\n\\begin{proof}\nSince $\\d$ and $N_1$ commute,\n\\[ \\d P_{a,b} \\subseteq \\ker(N_1^{a+1})\\cap H_{a-1,b-1} = \\bigoplus_{r\\geq 0} \\ N_2^r P_{a-1,b-1+2r} \\oplus N_1N_2^r P_{a+1,b-1+2r}. \\]\nNotice that this is true even if $a=0$ or $b=0$. We conclude the proof by combining this result with the symmetric result for $N_2$.\n\\end{proof}\n\nUsing this proposition, if $x\\in P_{a,b}$, then for $i,j\\in\\{0,1\\}$, we define $x_{i,j}\\in P_{a-1+2i,b-1+2j}$ such that\n\\[ \\d x = \\sum_{i,j\\in\\{0,1\\}}N_1^iN_2^jx_{i,j}. \\]\nSince $\\d$ commutes with $N_1$ and $N_2$, more generally, for any pair of integers $r,s\\geq0$, we get\n\\[ \\d N_1^r N_2^s x = \\sum_{i,j\\in\\{0,1\\}}N_1^{r+i}N_2^{s+j}x_{i,j}. \\]\n\nWith this notation, the fact that $\\d^2=0$ can be reformulated as follows.\n\n\\begin{prop} \\label{prop:dd=0}\nWith the above notations, for any pair $k,l\\in\\{0,1,2\\}$,\n\\[ N_1^{\\epsilon_1}N_2^{\\epsilon_2}\\sum_{i,j,i',j'\\in\\{0,1\\} \\\\ i+i'=k \\\\ j+j'=l}(x_{i,j})_{i',j'}=0. \\]\nwhere,\n\\[ \\epsilon_1 = \\begin{cases}\n  1 & \\text{if $a=0$ and $k=1$} \\\\\n  0 & \\text{otherwise,}\n\\end{cases} \\quad\\text{ and }\\quad\n\\epsilon_2 = \\begin{cases}\n  1 & \\text{if $b=0$ and $l=1$} \\\\\n  0 & \\text{otherwise.}\n\\end{cases} \\]\n\\end{prop}\n\n\\begin{proof}\nFrom the previous discussion, we get that\n\\[ \\d^2 x \\in \\bigoplus_{k,l\\in\\{0,1,2\\}} N_1^kN_2^lP_{a-2+2k,b-2+2l}. \\]\nMoreover, the piece corresponding to a fixed $k$ and $l$ is\n\\[ \\sum_{i,j,i',j'\\in\\{0,1\\} \\\\ i+i'=k \\\\ j+j=l}N_1^kN_2^l(x_{i,j})_{i',j'}. \\]\nSince, $\\d^2=0$, this last sum is zero:\n\\[ N_1^kN_2^l\\sum_{i,j,i',j'\\in\\{0,1\\} \\\\ i+i'=k \\\\ j+j=l}(x_{i,j})_{i',j'}=0. \\]\n\nWe know that, for any $b'$, $N_1^k\\rest{P_{a-2+2k,b'}}$ is injective if $k\\leq a-2+2k$. This is also true if $a-2+2k<0$ because, in this case, the domain is trivial. In both cases, we can remove $N_1^k$ in the above equation. The last case is\n\\[ 0\\leq a-2+2k < k. \\]\nThis implies $k=1$ and $a=0$, in which case we cannot a priori remove the $N_1^k=N_1$ in the previous equation. This concludes the proof of the proposition.\n\\end{proof}\n\n\n\n\\subsection{The Laplace operator $\\Lap$} The \\emph{Laplace operator} $\\Lap\\colon H_{\\bul,\\bul}\\to H_{\\bul,\\bul}$ is defined by\n\\[ \\Lap := \\d \\dd + \\dd \\d \\]\nwhere $\\dd\\colon H_{\\bul,\\bul}\\to H_{\\bul,\\bul}$ denotes the \\emph{codifferential} of bidegree $(1,1)$ defined by\n\\[ \\dd:= -\\w^{-1} \\d \\w. \\]\n\nNote that $\\Lap$ is of bi-degree $(0,0)$. The following summarizes basic properties one expect from the Laplace operator and its corresponding Hodge decomposition.\n\\begin{prop} \\label{prop:Lap}\nWe have the following properties.\n\\begin{itemize}\n\\item $\\dd$ is the adjoint of $\\d$ with respect to $\\phi$.\n\\item $\\Lap$ is symmetric for $\\phi$.\n\\item On each graded piece $H_{a,b}$, we have the following orthogonal decomposition for $\\phi$\n\\[ \\ker(\\d)\\cap H_{a,b} = \\ker(\\Lap)\\cap H_{a,b} \\wooplus \\Im(\\d)\\cap H_{a,b}. \\]\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}To get the first point, note that for any $x,y\\in H_{\\bul, \\bul}$, we have\n\\[ \\phi(x,\\dd y)=-\\psi(x,\\w \\w^{-1}\\d\\w y)=\\psi(\\d x,\\w y)=\\phi(\\d x, y). \\]\n\nIt follows that the adjoint of $\\Lap=\\d\\dd+\\dd\\d$ is $\\Lap$ itself.\n\n\\medskip\n\nLast point follows directly from the first two statements, as usual. Denote by $V:=(H_{a,b},\\phi)$ the vector space $H_{a,b}$ endowed with the inner product $\\phi$. In order to simplify the notations, we will write $\\Im(\\d)$ instead of $V\\cap\\Im(\\d)$, and similarly for other intersections with $V$. We have to show that\n\\[ \\ker(\\d)=\\ker(\\Lap)\\ooplus\\Im(\\d).\\]\nFrom the decomposition $V=\\Im(\\d)^\\perp \\oplus \\Im(\\d)$, and the inclusion $\\Im(\\d)\\subseteq\\ker(\\d)$, we get\n\\[ \\ker(\\d)=\\Im(\\d)^\\perp\\cap\\ker(\\d) \\wooplus \\Im(\\d). \\]\n\nBy adjunction, we get $\\Im(\\d)^\\perp=\\ker(\\dd)$. To conclude, note that $\\ker(\\Lap)=\\ker(\\d)\\cap\\ker(\\dd)$. Indeed, if $x\\in\\ker(\\Lap)$, then\n\\[ 0 = \\phi(x,\\Lap x) = \\phi(x, \\d\\dd x)+\\phi(x, \\dd\\d x) = \\phi(\\dd x, \\dd x)+\\phi(\\d x, \\d x), \\]\nwhich, by positivity of $\\phi$, implies that $\\dd x=\\d x=0$, i.e., $x\\in\\ker(\\d)\\cap\\ker(\\dd)$. The other inclusion $\\ker(\\d)\\cap\\ker(\\dd) \\subseteq \\ker(\\Lap)$ trivially holds.  \\qedhere\n\\end{proof}\n\n\\begin{prop}\\label{prop:commutativity} The Laplace operator commutes with $N_1$ and $N_2$, i.e., we have $[\\Lap,N_1]=[\\Lap,N_2]=0$.\n\\end{prop}\nThe proof of this theorem is given in Section \\ref{subsec:Laplacian_commute}.\n\n\n\n\\subsection{Polarized Hodge-Lefschetz structure on the cohomology of $H_{\\bul,\\bul}$}\n\nIn this section, we show that, when passing to the cohomology, the operators $N_1$, $N_2$ and the polarization $\\psi$ induce a polarized Hodge-Lefschetz structure on the cohomology groups\n\\[ L_{a,b} := \\frac{\\ker(\\d\\colon H_{a,b}\\to H_{a-1,b-1})}{\\Im(\\d\\colon H_{a+1,b+1}\\to H_{a,b})}. \\]\nBy this we mean the following.\n\nFirst, since $[\\d, N_1] =0$ and $[\\d,N_2]=0$, we get induced maps\n\\[ N_1 \\colon L_{a,b} \\to L_{a-1, b} \\quad\\text{ and }\\quad N_2\\colon L_{a,b} \\to L_{a, b-2}. \\]\n\nSecond, since $\\psi(\\d\\,\\cdot\\,,\\cdot\\,)=-\\psi(\\,\\cdot\\,,\\d\\,\\cdot\\,)$, we get a pairing\n\\[ \\psi\\colon L_{\\bul,\\bul}\\times L_{\\bul,\\bul} \\to \\mathbb{R}. \\]\n\nThen, we will show that\n\\begin{thm} \\label{thm:differential_HL-structure}\nFor the corresponding induced map, $(L_{\\bul,\\bul}, N_1, N_2, \\phi)$ is a polarized HL-structure. Moreover, the decomposition by primitive parts is induced by the corresponding decomposition on $H_{\\bul,\\bul}$.\n\\end{thm}\n\n\\begin{proof}\nBy Proposition \\ref{prop:Lap}, the kernel of the Laplace operator gives a section of the cohomology $L_{\\bul,\\bul}$. Moreover, since the Laplacian commutes with $N_1$ and $N_2$, both $N_1$ and $N_2$ preserve the kernel of the Laplacian. Consequently, it suffices to prove that $\\ker(\\Lap)$ is a polarized HL-substructure of $H_{\\bul,\\bul}$. By abuse of notation, we denote by $L_{a,b}$ the kernel $\\ker(\\Lap\\colon H_{a,b}\\to H_{a,b})$.\n\nClearly, $N_1$ and $N_2$ commutes. Let $a\\geq 0$ and $b\\in\\mathbb{Z}$. Let us prove that $N_1^a$ induces an isomorphism $L_{a,b}\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} L_{-a,b}$. Since $\\Lap$ is symmetric for $\\phi$,\n\\[ H_{a,b}=\\ker(\\Lap)\\cap H_{a,b} \\wooplus \\Im(\\Lap)\\cap H_{a,b}. \\]\nSince $N_1$ preserves $\\ker(\\Lap)$ and $\\Im(\\Lap)$, the isomorphism $N_1^a\\colon H_{a,b}\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} H_{-a,b}$ induces isomorphisms for both parts $\\ker(\\Lap)$ and $\\Im(\\Lap)$. In particular, we get an isomorphism $N_1^a\\colon L_{a,b}\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} L_{-a,b}$. We use the symmetric argument for $N_2$.\n\nThe primitive part of $L_{a,b}$ is, by definition,\n\\[ L_{a,b}\\cap\\ker(N_1^{a+1})\\cap\\ker(N_2^{b+1})=L_{a,b}\\cap P_{a,b}. \\]\nThus, the decomposition by primitive parts is the one induced by the decomposition of $H_{a,b}$. Finally, the properties about $\\psi$ on $L_{a,b}$ comes from the analogous properties on $H_{a,b}$.\n\\end{proof}\n\n\n\n\\subsection{Commutative property of the Laplacian}\n\\label{subsec:Laplacian_commute}\n\n{\n\\renewcommand{\\r}{{r}}\n\\renewcommand{\\a}{{ a}}\n\\newcommand{{0}}{{0}}\n\nIn this section, we prove Proposition~\\ref{prop:commutativity}, that $[\\Lap,N_1]=[\\Lap,N_2]=0$.\n\n\\medskip\n\nLet us start by justifying our definition of the operator $\\w$.\n\\begin{remark} We feel necessary to provide an explanation of why we are not using the simpler more natural operator $\\~\\w$ which is defined by taking values for $a,b,r,s\\geq0$ and $x\\in P_{a,b}$ given by\n\\[ \\~\\w N_1^rN_2^sx=(-1)^{r+s}N_1^{a-r}N_2^{b-s}x, \\]\nwhere negative powers of $N_1$ and $N_2$ are considered to be zero.\n\nWith this operator, the bilinear form $\\~\\phi:=\\psi(\\,\\cdot\\,,\\~\\w\\cdot\\,)$ will be still positive definite, and we can define the corresponding Laplace operator $\\~\\Lap$. However, this Laplacian will not commute with $N_1$ and $N_2$ in general. In order to explain where the problem comes from, let us try to proceed with a natural, but \\emph{wrong}, proof that this Laplacian commutes $N_1$. More precisely, setting $\\~\\dd = -\\~\\w^{-1}\\d\\~\\w$ the adjoint of $\\d$, let us try to prove that $[\\~\\dd,N_1]=0$. Assume $0\\leq r\\leq a$ and $0\\leq s\\leq b$. Then we can write\n\\begin{align*}\n\\~\\dd N_1^rN_2^sx\n  &= -\\~\\w^{-1}\\d\\~\\w N_1^rN_2^sx \\\\\n  &= (-1)^{r+s+1}\\~\\w^{-1}\\d N_1^{a-r}N_2^{b-s}x \\\\\n  &= \\sum_{i,j\\in\\{0,1\\}}(-1)^{r+s+1}\\~\\w^{-1}N_1^{a-r+i}N_2^{b-s+j}x_{i,j}.\n\\end{align*}\nWe recall that $x_{i,j}\\in P_{a-1+2i,b-1+2j}$. Notice that $a-1+2i-(a-r+i)=r-1+i$. Thus,\n\\begin{align*}\n\\~\\dd N_1^rN_2^sx\n  &= \\sum_{i,j\\in\\{0,1\\}}(-1)^{r+s+1}(-1)^{r-1+i+s-1+j}N_1^{r-1+i}N_2^{s-1+j}x_{i,j} \\\\\n  &= \\sum_{i,j\\in\\{0,1\\}}(-1)^{i+j+1}N_1^{r-1+i}N_2^{s-1+j}x_{i,j}.\n\\end{align*}\nThen, $\\~\\dd N_1(N_1^rN_2^sx)$ can be obtained replacing $r$ by $r+1$ and similarly for $N_1\\~\\dd(N_1^rN_2^sx)$. Thus, we would conclude that $N_1$ and $\\~\\dd$ commute!\n\nWhere is the mistake? First, notice that the formula for the derivation can only be applied if $a-r$ and $b-s$ are both non-negative. Thus, if $r=a$, we cannot just replace $r$ by $r+1$ to compute $\\~\\dd N_1(N_1^rN_2^sx)$. Second, notice that $0=N_1N_1^{-1}\\neq N_1^{0}$. If $r=0$ and $i=0$, we get $r-1+i=-1$. Multiplying the corresponding term by $N_1$ is not equivalent to simply replacing $r$ by $r+1$. Notice moreover that we did not use the condition $\\d^2=0$ at any moment. Thus, roughly speaking, everything goes well except on the boundaries $r=a, s=b, r=0$ and $s=0$. These problems still exist for the operator $\\~\\Lap$, which justify why $\\~\\Lap$ does not commute in general with $N_i$.\n\n\\medskip\n\nThe solution to remedy these boundary issues is to modify $\\~\\w$ in such a way that some zero factors on the boundary solve the above evoked problem: if the term $N_1^{r-1+i}N_2^{s-1+j}x_{i,j}$ is multiplied by some zero factor due to the new operator when $r=0$ and $i=0$, then multiplying by $N_1$ corresponds to replacing $r$ by $r+1$. To find the right form of the Laplacian, one can use tools of representation theory, where Lie brackets can be used to study commutation properties. This is what is done in \\cites{GNA90, SabSch}. These proofs however take place in a slightly different context. For example, our differential is skew symmetric, and not symmetric, and we work in the real instead of the complex world. Nevertheless, one should be able to adapt their proofs to our setting. However, instead of doing this, we decided to provide a completely elementary proof. The arguments which follow are somehow technical but no knowledge in representation theory is needed. \\end{remark}\n\n\\medskip\n\nWe now begin the proof of the commutativity of the Laplacian $\\Lap$ with the operators $N_1$ and $N_2$.\n\n\\medskip\n\n\\paragraph{{\\bf Convention.}} In order to simplify the calculations, we avoid to write symmetric factors by using the following notations. If $A,B,C$ are some sets, then we follow the following rules to extend the operations between these sets coordinatewise to operations involving their Cartesian products $A^2,B^2$ and $C^2$: if $\\zeta\\colon A\\to C$ and $\\theta\\colon A\\times B\\to C$ are some operations, then we extend them as follows.\n\\[ \\begin{array}{rrrll}\n\\zeta\\colon  & A^2\\to C^2, \\qquad             & (a_1,a_2)                       & \\mapsto (\\zeta(a_1), \\zeta(a_2)),                      \\\\\n\\theta\\colon & A \\times B^2 \\to C^2, \\qquad   & \\bigl(a,(b_1,b_2)\\bigr)         & \\mapsto (\\theta(a,b_1), \\theta(a,b_2)), \\text{ and } \\\\\n\\theta\\colon & A^2 \\times B^2 \\to C^2, \\qquad & \\bigl((a_1,a_2),(b_1,b_2)\\bigr) & \\mapsto (\\theta(a_1, b_1), \\theta(a_2, b_2)).\n\\end{array} \\]\nMoreover, if $A$ is endowed with a product structure, we define $\\m\\colon A^2\\to A$ by $\\m(a_1,a_2)=a_1a_2$.\n\n\\begin{example}\nLet us give some examples. Let $a=(a_1, a_2)$ and $r=(r_1,r_2)$ for integers $a_1,a_2,r_1,r_2\\geq 0$ with $r_1\\leq a_1$ and $r_2\\leq a_2$, and let $x\\in P_{a,b}$. With our convention, we get $1+a = 1+(a_1, a_2) = (1+a_1, 1+a_2)$, $(-1)^{a} = (-1)^{(a_1, a_2)} = \\bigl((-1)^{a_1}, (-1)^{a_2}\\bigr)$, and $\\m\\bigl((-1)^{1+a}\\bigr) =\\m\\bigl((-1)^{a}\\bigr) $.\n\nLet now $N=(N_1, N_2)$. The formula\n\\[ \\w N_1^{r_1}N_2^{r_2}x=(-1)^{r_1+r_2}\\frac{r_1!}{(a_1-r_1)!}\\frac{r_2!}{(a_2-r_2)!}N_1^{a_1-r_1}N_2^{a_2-r_2}x, \\]\ncan be rewritten with our notation as follows\n\\[ \\w \\m(N^\\r)x=\\m\\Bigl((-1)^r\\frac{r!}{(a-r)!}N^{a-r}\\Bigr)x. \\]\nIn the same way, we recall that\n\\[ \\d N_1^{r_1} N_2^{r_2} x = \\sum_{i=(i_1,i_2)\\in\\{0,1\\}^2}N_1^{r_1+i_1}N_2^{r_2+i_2}x_{i_1,i_2}, \\]\nif $r_1$ and $r_2$ are \\emph{non-negative}, where $x_{i_1,i_2}\\in P_{a_1-1+2i_1,a_2-1+2i_2}$. With our notations, this formula reads\n\\[ \\d \\m(N^r) x = \\sum_{i\\in\\{0,1\\}^2}\\m(N^{r+i})x_i. \\]\n\\end{example}\n\nWe further use the following conventions. By $(a_1,a_2)\\geq(b_1,b_2)$ we mean $a_1\\geq b_1$ and $a_2\\geq b_2$. Moreover, $0^0=1$ and $N_1^r=N_2^r=0$ if $r$ is negative. Notice that $N_1N_1^{-1}\\neq N_1^0$, thus we have to be cautious with negative exponents. This is the main difficulty in the proof, besides the exception $k=1$ and $a=0$ in Proposition \\ref{prop:dd=0}.\n\n\\medskip\n\nLet $a=(a_1,a_2)\\geq 0$. Let $r=(r_1,r_2)\\in\\mathbb{Z}^2$ with $0\\leq r\\leq a$. Let $x\\in P_a=P_{a_1,a_2}$ and $y=N_1^{r_1}N_2^{r_2}x$. Set $N=(N_1,N_2)$. We recall that\n\\[ \\Lap=-(\\d\\w^{-1}\\d\\w+\\w^{-1}\\d\\w\\d) \\]\n\nWe now explicit $\\d \\w^{-1} \\d y$. We omit ranges of indices in the sums which follow if they are clear from the context.\n\\begin{align*}\n\\d y\n  &= \\sum_{i\\in\\{0,1\\}^2}\\m\\bigl(N^{r+i}\\bigr) x_i. \\\\\n\\w^{-1} \\d y\n  &= \\sum_{i \\\\ r+i\\leq a-1+2i}\\m\\Bigl((-1)^{a-1+2i}(-1)^{r+i}\\frac{(r+i)!}{((a-1+2i)-(r+i))!}N^{(a-1+2i)-(r+i)}\\Bigr) x_i, \\\\\n  &= \\sum_{i \\\\ a-r-1+i\\geq0}\\m\\Bigl((-1)^{a+r+i+1}\\frac{(r+i)!}{(a-r-1+i)!}N^{a-r-1+i}\\Bigr) x_i. \\\\\n\\d \\w^{-1} \\d y\n  &= \\sum_{i,i'\\in\\{0,1\\}^2 \\\\ a-r-1+i\\geq0} \\m\\Bigl((-1)^{a+r+i}\\frac{(r+i)!}{(a-r-1+i)!}N^{a-r-1+i+i'}\\Bigr) (x_i)_{i'}.\n\\end{align*}\nwhere we used the fact that $\\m\\bigl((-1)^{a+r+i+1}\\bigr)=\\m\\bigl((-1)^{a+r+i}\\bigr)$ (since $\\m\\bigl((-1)^{(1,1)}\\bigr) =1$).\n\n\\medskip\n\nWe can compute in the same way $\\d \\w \\d y$, in which case, using\n\\[ \\m\\bigl((-1)^{a-1+2i}\\bigr)=\\m\\bigl((-1)^a\\bigr), \\]\nwe get\n\\[ \\d \\w \\d y=\\m\\bigl((-1)^a\\bigr)\\d \\w^{-1} \\d y. \\]\n\n\\medskip\n\nTo get $\\d \\w^{-1} \\d \\w y$, it suffices to multiply by the good factor and to replace $r$ by $a-r$:\n\\begin{align*}\n\\d \\w^{-1} \\d \\w y\n  &= \\m\\Bigl((-1)^{r}\\frac{r!}{(a-r)!}\\Bigr) \\sum_{i,i'\\in\\{0,1\\}^2 \\\\ r+i-1\\geq0}\\m\\bigl((-1)^{r+i}\\frac{(a-r+i)!}{(r-1+i)!}N^{r-1+i+i'}\\bigr) (x_i)_{i'} \\\\\n  &= \\sum_{i,i' \\\\ r+i-1\\geq0}\\m\\bigl((-1)^{i}\\frac{r!}{(r-1+i)!}\\frac{(a-r+i)!}{(a-r)!}N^{r-1+i+i'}\\bigr) (x_i)_{i'} \\\\\n  &= \\sum_{i,i' \\\\ r+i-1\\geq0}\\m\\Bigl((-1)^{i}r^{1-i}(a-r+1)^i N^{r-1+i+i'}\\Bigr) (x_i)_{i'}.\n\\end{align*}\nNote that in the last sum we can remove the condition $r+i-1\\geq0$. Indeed, if, for instance, $r_1+i_1-1<0$, then, since $r_1, i_1$ are non-negative, we should have $r_1=i_1=0$, and the factor $r_1^{1-i_1}$ is automatically zero.\n\n\\medskip\n\nNow, since\n\\[ (a-2+2i+2i')-(a-r-1+i+i')=r-1+i+i', \\]\nwe get\n\\begin{align*}\n\\w^{-1} &\\d \\w \\d y \\\\\n  &= \\hspace{-5mm}\\sum_{i,i' \\\\ a-r+i-1\\geq0}\\hspace{-5mm}\\m\\Bigl((-1)^{r+i}\\frac{(r+i)!}{(a-r-1+i)!}(-1)^{r-1+i+i'}\\frac{(a-r-1+i+i')!}{(r-1+i+i')!}N^{r-1+i+i'}\\Bigr) (x_i)_{i'} \\\\\n  &= \\hspace{-5mm}\\sum_{i,i' \\\\ a-r+i-1\\geq0}\\hspace{-5mm}\\m\\Bigl((-1)^{i'}(r+i)^{1-i'}(a-r+i)^{i'}N^{r-1+i+i'}\\Bigr) (x_i)_{i'}.\n\\end{align*}\nIn the last sum, we can again remove the condition $a-r+i-1\\geq0$. Indeed, if, for instance, $a_1-r_1+i_1-1<0$, then since $r_1\\leq a_1$ and $i_1 \\in\\{0,1\\}$, we get $a_1=r_1$ and $i_1=0$. Then, either $i'_1=1$ and $(a_1-r_1+i_1)^{i'_1}=0$, or $i'_1=0$ and\n\\[ N_1^{r_1-1+i_1+i'_1}(x_i)_{i'}=N_1^{a_1-1}(x_i)_{i'}=0, \\]\nbecause\n\\[ (x_i)_{i'}=(x_{0,i_2})_{0,i'_2}\\in P_{a_1-2,a_2-2+i_2+i'_2}\\subseteq\\ker(N_1^{a_1-1}). \\]\n\n\\smallskip\nIn any cases, we get\n\\begin{align*}\n-\\Lap y   &= \\sum_{i,i'} \\Bigl(\\m\\bigl((-1)^{i}r^{1-i}(a-r+1)^i\\bigr) + \\m\\bigl((-1)^{i'}(r+i)^{1-i'}(a-r+i)^{i'}\\bigr)\\Bigr) \\m(N^{r-1+i+i'})(x_i)_{i'} \\\\\n  &= \\sum_{i,i'} \\Bigl(\\m\\bigl(r^{1-i}(r-a-1)^i\\bigr) + \\m\\bigl((r+i)^{1-i'}(r-a-i)^{i'}\\bigr)\\Bigr) \\m(N^{r-1+i+i'})(x_i)_{i'} \\\\\n  &= \\sum_{i,i'} \\Bigl(\\m\\bigl(r+0^{1-i}(-a-1)^i\\bigr) + \\m\\bigl(r+i^{1-i'}(-a-i)^{i'}\\bigr)\\Bigr) \\m(N^{r-1+i+i'})(x_i)_{i'}.\n\\end{align*}\n\n\\medskip\n\nFor a fixed $a$, let $\\kappa\\colon \\{0,1,2\\}^2\\to\\mathbb{Z}^2$ be the function defined by taking for every $i,i'\\in\\{0,1\\}^2$, the value\n\\[ \\kappa(i+i'):=0^{1-i}(-a-1)^i+i^{1-i'}(-a-i)^{i'}. \\]\nOne can verify that the function is well-defined, i.e., if $i+i' = j+j$ for $i,i',j,j' \\in \\{0,1\\}^2$ then\n\\[0^{1-i}(-a-1)^i+i^{1-i'}(-a-i)^{i'} = 0^{1-j}(-a-1)^j+j^{1-j'}(-a-j)^{j'}.\\]\nThe only nontrivial case is when $i_1+i'_1=1$ or when $i_2+i'_2=1$. For instance, if $i_1=0, i'_1=1$ and $j_1=1, j_1' =0$, then we get $\\kappa(i+i')_1=0-a_1=-a_1$ which is equal to $\\kappa(j+j')_1$.\n\n\\medskip\n\nWe also define $C\\colon \\{0,1\\}^2\\to\\mathbb{Z}$ by\n\\[ C(i,i'):=\\m\\bigl(0^{1-i}(-a-1)^i\\bigr) + \\m\\bigl(i^{1-i'}(-a-i)^{i'}\\bigr).\\]\n\n\\medskip\n\nWe will use the following specific values later:\n\\begin{gather*}\n\\kappa\\bigl((1,k_2)\\big)_1=-a_1, \\\\\nC\\bigl((0,i_2),(0,i'_2)\\bigr)=0, \\text{ and } \\\\\nC\\bigl((1,i_2),(1,i'_2)\\bigr)=(-a_1-1)\\kappa\\bigl((2, i_2+i'_2)\\bigr)_2.\n\\end{gather*}\n\n\\medskip\n\nUsing these terminology, we can now write\n\\begin{align*}\n\\Lap y\n  &= -\\sum_{i,i'} (2r_1r_2+\\kappa(i+i')_2r_1+\\kappa(i+i')_1r_2+C(i,i')) \\m(N^{r-1+i+i'})(x_i)_{i'}.\n\\end{align*}\n\nSet\n\\begin{equation} \\label{eqn:Lap'}\n\\Lap' y := -\\sum_{i,i'}C(i,i')\\m(N^{r-1+i+i'})(x_i)_{i'}.\n\\end{equation}\n\nWe claim that $\\Lap y=\\Lap' y$. Indeed, we can write\n\\begin{align*}\n(\\Lap'-\\Lap) y\n  &= \\sum_{k\\in\\{0,1,2\\}^2} \\Bigl(2r_1r_2+\\kappa(k)_2r_1+\\kappa(k)_1r_2\\Bigr)\\m(N^{r-1+k}) \\sum_{i,i'\\in\\{0,1\\}^2 \\\\ i+i'=k} (x_i)_{i'}.\n\\end{align*}\nSince $\\d^2=0$, using Proposition \\ref{prop:dd=0}, we get\n\\[\\m(N^{r-1+k}) \\sum_{i,i'\\in\\{0,1\\}^2 \\\\ i+i'=k} (x_i)_{i'} = 0\\]\nin the above sum unless either, for $k_1=1$ and $a_1=0$, or for $k_2=1$ and $a_2=0$.\n\nIn the first case $k_1=1$ and $a_1=0$, we should have $r_1=0$ and the remaining factor is $\\kappa((1,k_2))_1r_2$. We have seen that $\\kappa((1,k_2))_1=-a_1$. Thus, here, the factor is zero and there is no contribution in the sum. The same reasoning applies by symmetry to the case $k_2=1$ and $a_2=0$. Therefore, in any case, we get $\\Lap y=\\Lap' y$ for any $y$, and the claim follows.\n\n\\medskip\n\nFinally, we prove that $\\Lap'$ commutes with $N_1$. Recall that $y=N_1^{r_1} N_2^{r_2} x$ and $x\\in P_{a_1,a_2}$. We divide the proof into three cases, depending on whether $r_1=0, a_1$ or $0<r_1 <a_1$\n\n\\medskip\n\n\\begin{itemize}[leftmargin=0pt, itemindent=1.5em, labelsep=.5em]\n\\item If $0=r_1<a_1$, we can compute $\\Lap'N_1y$ by replacing $r_1$ by $r_1+1$ in Equation \\eqref{eqn:Lap'}. For $N_1\\Lap'y$, this remains true except for the terms with $i_1=i'_1=0$. Indeed,\n\\[ N_1N_1^{r_1-1+0+0}=0\\neq N_1^{r_1+1-1+0+0}=\\id. \\]\nLuckily, for those terms, the factor is $C((0,i_2),(0,i'_2))$, which is zero. Thus, in fact, we can also do the replacement for those terms. Hence, $\\Lap'N_1y=N_1\\Lap'y$.\n\n\\medskip\n\n\\item If $0<r_1<a_1$, then we can compute $\\Lap'N_1y$ by replacing $r_1$ by $r_1+1$, and $N_1\\Lap'y$ in the same way. Thus, they are equal.\n\n\\medskip\n\n\\item Finally, if $r_1=a_1$, since we used the hypothesis $r_1\\leq a_1$ in all the computations leading to the expression of $\\Lap'$, we cannot apply those computations to $N_1y$. However, in this case, we have $N_1y=0$. Thus, we just need to prove that $N_1\\Lap'y=0$. This can be done thanks to the defining Equation \\eqref{eqn:Lap'}.\n\nIf $i_1+i'_1\\leq 1$, then $1+r_1-1+i_1+i'_1\\geq a_1-2+2(i_1+i'_1)+1$. Since $(x_i)_{i'}\\in P_{a_1-2+2(i_1+i'_1), b'}$ for some $b'$, we get\n\\[ C(i,i')N_1\\m(N^{r-1+i+i'})(x_i)_{i'}=0. \\]\nThe only remaining terms are those where $i_1=i'_1=1$. For these terms, we have $C(i,i')=(-a_1-1)\\kappa(i+i')_2$. Thus,\n\\[ N_1\\Lap'y=(a_1+1)N_1\\sum_{k_2\\in\\{0,1,2\\}}\\kappa((2,k_2))_2\\m(N^{r-1+i+i'})\\sum_{i_2,i'_2\\in\\{0,1\\}^2 \\\\ i_2+i'_2=k_2} (x_{1,i_2})_{1,i'_2}. \\]\nAs in the first case, we apply Proposition \\ref{prop:dd=0} to infer that all the terms in this sum are zero unless for $k_2=1$ and $a_2=0$. For these values of $k_2$ and $a_2$, we use $\\kappa((2,1))_2=-a_2=0$ to conclude.\n\\end{itemize}\n\n\\medskip\n\nIn all the cases, we see that $N_1$ commutes with $\\Lap'$. Since we proved $\\Lap=\\Lap'$, we conclude with $[\\Lap,N_1]=0$, and by symmetry, $[\\Lap,N_2]=0$, as required.\n}\n\n\n\n\n\n\n\n\n\\section{Projective bundle theorem}\n\\label{sec:projective_bundle_theorem}\n\nLet $\\mathfrak X$ be a polyhedral space in $V = N_\\mathbb{R}$. We define the \\emph{asymptotic cone} of $\\mathfrak X$ which we denote by $\\mathfrak X_\\infty$ to be the limit set of the rescaled spaces $\\mathfrak X\/t$ when $t$ goes to $+\\infty$. Exceptionally, the terminology cone here might refer to a non-necessarily convex cone. Note that, if $X$ is any polyhedral complex structure on $\\mathfrak X$, then $\\mathfrak X_\\infty=\\supp{X_\\infty}$.\n\n\\medskip\n\nLet $\\mathscr Y$ be a smooth polyhedral space in $V = N_\\mathbb{R}$. Let $\\Delta$ be a unimodular fan subdividing $\\mathscr Y_\\infty$. Let $\\mathfrak X$ be the closure of $\\mathscr Y$ in $\\TP_\\Delta$. For each cone $\\eta \\in \\Delta$, we define $D^\\eta$ as the closure of the intersection $\\mathfrak X \\cap \\TP_\\Delta^\\eta$.\n\n\\medskip\n\nUsing these notations, we see in particular that we have\n\\begin{itemize}\n\\item $D^{\\conezero} = \\mathfrak X$.\n\\item $D^\\eta$ are all smooth of dimension $d-\\dims{\\eta}$.\n\\item We have \\[D^\\eta = \\bigcap_{\\substack{\\varrho \\prec \\eta \\\\ \\dims{\\varrho}=1}} D^\\varrho.\\]\n\\end{itemize}\nIn other words, the boundary $D = \\mathfrak X \\setminus \\mathscr Y$ is a \\emph{simple normal crossing divisor} in $\\mathfrak X$.\n\n\n\n\\subsection{Cycle class associated to the cones in $\\Delta$} In this section, we recall how to associate elements of the tropical cohomology groups to the strata $D^\\eta$ defined in the preceding paragraph. So we fix the notations as above. Let $\\Delta$ be a unimodular fan structure on the asymptotic cone $\\mathscr Y_\\infty$, and take $\\eta \\in \\Delta$. The stratum $D^\\eta \\subset \\mathfrak X$ defines then an element $\\class(D^\\eta)$ of $H^{\\dims\\eta,\\dims\\eta}_\\trop(\\mathfrak X)$ obtained as follows.\n\n\\medskip\n\nLet $d$ be the dimension of $\\mathscr Y$. From the inclusion $D^\\eta \\subseteq \\mathfrak X$, we get a map\n\\[\n\\begin{tikzcd}[row sep=tiny]\nH_{d-\\dims\\eta, d-\\dims\\eta} (D^\\eta) \\rar & H_{d-\\dims \\eta, d-\\dims\\eta} (\\mathfrak X) \\rar[\"\\sim\"] &\n\\Bigl(H^{d -\\dims \\eta, d-\\dims \\eta} (\\mathfrak X) \\Bigr)^\\dual \\rar[\"\\sim\"]& H^{\\dims\\eta,\\dims\\eta}(\\mathfrak X)\\\\\n{[D^{\\eta}]}\\arrow[mapsto]{rrr} &&&\\class(D^\\eta),\n\\end{tikzcd}\n\\]\ni.e., $\\class(D^\\eta)$ is the image of $[D^\\eta] \\in H_{d-\\dims\\eta, d-\\dims\\eta} (D^\\eta) $ in $H^{\\dims\\eta,\\dims\\eta}(\\mathfrak X)$ by the composition of above maps. Note that the last isomorphism in the above chain of maps follows from the Poincar\\'e duality for $\\mathfrak X$.\n\n\n\n\\subsection{Explicit description of the divisor classes associated to the rays} \\label{sec:cl_D'}\n\nConsider now the situation where $\\mathfrak X$ comes with a unimodular polyhedral complex structure $X$ with $X_\\infty =\\Delta$. Moreover, we consider a ray $\\rho$ of $\\Delta$  and its associated stratum $D^\\rho$ and its cycle class $\\class(D^\\rho) \\in H^{1,1}_\\trop(\\mathfrak X)$. In this case, there is a nice representative $c = (c_\\zeta)_{\\substack{\\zeta\\in X \\\\ \\dims \\zeta=1}}$ in $C^{1,1}_\\trop(X)$ of $\\class(D^\\rho)$, with $c_\\zeta \\in \\SF^1(\\zeta)$, as we explain now. For details of this construction, we refer to~\\cite{AP}.\n\n\\medskip\n\nFirst we define the elements $\\zeta \\in X$ of dimension one which are \\emph{parallel to $\\rho$}.\n\nSo let $\\zeta$ be a one dimensional face of $X$ and denote by $\\tau \\in \\Delta$ the sedentarity of $\\zeta$.\nWe say \\emph{$\\zeta$ is parallel to $\\rho$} if the following two properties hold:\n\\begin{enumerate}\n\\item $\\sigma:=\\rho+\\tau$ is a cone  of $\\Delta$ which properly contains $\\tau$; and\n\\item $\\zeta^\\tau=\\sigma^\\tau_\\infty$.\n\\end{enumerate}\nHere $\\sigma^\\tau_\\infty$ is $\\sigma\/\\tau$ seen in the stratum $N^\\tau_{\\infty, \\mathbb{R}}$ of $\\TP_\\Delta$, and $\\zeta^\\tau$ is the intersection $\\zeta \\cap N^\\tau_{\\infty, \\mathbb{R}}$.\n\n\\medskip\n\nWith this preliminary discussion, the element $c=(c_\\zeta)_{\\substack{\\zeta\\in X \\\\ \\dims \\zeta=1}}$ in $C^{1,1}_\\trop(X)$ is defined as follows.\n\n\\medskip\n\nFirst, consider a face $\\zeta\\in X$ which is not parallel to $\\rho$ and which has sedentarity $\\tau$ such that $\\tau$ does not contain $\\rho$ as a ray. For such $\\zeta$,  we set $c_\\zeta=0$.\n\n\\medskip\n\nSecond, consider a one dimensional cone $\\zeta$ of $X$ which is parallel to $\\rho$. Let $\\tau$ be the sedentarity of $\\zeta$ and choose a linear form $\\phi_\\zeta$ on $N^\\tau_{\\infty,\\mathbb{R}}$ which takes value one on the primitive vector $\\e_{\\zeta_\\infty}$ of $\\zeta_\\infty$. The elements $\\phi_\\zeta$ can be naturally seen as an element of $\\SF^1(\\zeta)$. We denote this element by $c_\\zeta$.\n\n\\medskip\n\nAll the remaining $\\zeta$ in $X$ are those whose sedentarity contains $\\rho$ as a ray. In particular, they are not parallel to $\\rho$. Any such $\\zeta$ form a side of a unique quadrilateral in $X$ with two sides $\\zeta_1, \\zeta_2$ parallel to $\\rho$, and the last side $\\zeta_3$ which is neither parallel to $\\rho$ nor with a sedentarity $\\sed(\\zeta_3)$ containing $\\rho$. By the construction of the previous two cases, we have $c_{\\zeta_3} =0$. Moreover,  the difference $c_{\\zeta_1} - c_{\\zeta_2}$ naturally defines a linear form on $N^\\tau_{\\infty, \\mathbb{R}}$. We define $c_{\\zeta} = \\pm(c_{\\zeta_1} - c_{\\zeta_2})$, for a convenient choice of sign that we omit to precise here.\n\n\\medskip\n\nThe element  $c=(c_\\zeta)_{\\substack{\\zeta\\in X \\\\ \\dims \\zeta=1}}$ in $C^{1,1}_\\trop(X)$ thereby defined is closed and its class in the cohomology $H^{1,1}_\\trop(X)$ coincides with $\\class(D^\\rho)$.\n\n\\medskip\n\nMoreover, to each face $\\zeta$ with sedentarity $\\conezero$ which is parallel to $\\rho$, we can associate the corresponding element $x_\\zeta$ of $A^1(\\zeta_\\f)=H^2(\\zeta_\\f)$. Note that $\\zeta_\\f$ is a vertex of $X_\\f$. Since $\\zeta_\\f$ is a vertex in $X_\\f$, we get an element of $\\ST_1^{0,2,0}\\subseteq\\ST_1^{0,2}$. By summing all the $x_\\zeta$, we get an element of $\\ST_1^{0,2}$ which is closed in the Steenbrink sequence, and which corresponds to $c$ via the isomorphism of Theorem \\ref{thm:steenbrink}.\n\n\\begin{remark} \\label{rem:explicit_function}\nWe have seen in Section \\ref{sec:projective_Kahler} how to associate an element of $\\ST_1^{0,2,0}$ to a piecewise linear function $f$ on $Y:=X^\\conezero$. Let us describe such a function giving the element $c$ considered in the preceding paragraph.\n\n\\medskip\n\nAssume $X$ is simplicial. Denote by $\\chi_\\rho$ the characteristic function of $\\rho$ on $\\Delta$ which takes value one on $\\e_\\rho$, value zero on each $\\e_\\varrho$ for all ray $\\varrho \\neq \\rho$ of $\\Delta$, and which is linear on each cone of $\\Delta$. Pick any linear form $\\e^\\dual_\\rho$ on $N_\\mathbb{R}$ which takes value $1$ on $\\e_\\rho$. Then $\\chi_\\rho$ can be naturally extended to $\\comp\\Delta$ as follows. For any cone $\\sigma\\in\\Delta$ not containing $\\rho$, we extend $\\chi_\\rho$ by continuity on $\\Delta^\\sigma$. For any cone $\\sigma\\in\\Delta$ containing $\\rho$, we define the extension $\\chi_\\rho$ on $\\Delta^\\sigma$ to be the continuous extension of $\\chi_\\rho-\\e^\\dual_\\rho$ to this space.\n\nThen $\\chi_\\rho$ naturally induces a piecewise linear function $\\~\\chi_\\rho$ on $X$ defined by $x\\mapsto\\chi_\\rho(x_\\infty)$. Moreover, the element of $\\ST_1^{0,2,0}$ associated to this function does not depend on the choice of $\\e^\\dual_\\rho$ and is equal to the element $c$ constructed above.\n\\end{remark}\n\n\n\n\\subsection{Statement of the theorem}\n\nLet now $\\sigma$ be a cone in $\\Delta$, and denote by $\\Delta'$ the fan obtained by the barycentric star subdivision of $\\sigma$ in $\\Delta$. Denote by $\\mathfrak X'$ the closure of $\\mathscr Y$ in $\\TP_{\\Delta'}$. By Theorem~\\ref{thm:smoothness-compact} we have the following.\n\\begin{prop}\nThe variety $\\mathfrak X'$ is smooth.\n\\end{prop}\n\nFor each cone $\\zeta$ in $\\Delta'$, we define similarly as in the preceding section, the corresponding closed stratum $D'^\\zeta$ of $\\mathfrak X'$. Let $\\rho$ be the new ray in $\\Sigma'$ obtained after the star subdivision of $\\sigma \\in \\Delta$.\n\nNote that $\\rho$ is the ray generated by the sum of the primitive vectors of the rays of $\\sigma$, i.e.,\n\\[\\rho = \\mathbb{R}_+\\bigl(\\sum_{\\varrho \\prec \\sigma \\\\ \\dims{\\varrho}=1} \\e_\\varrho\\bigr).\\]\nConsider the divisor $D'^\\rho \\subseteq \\mathfrak X'$ and its corresponding cycle class $\\class(D'^\\rho)$ of $H^{1,1}_\\trop(\\mathfrak X')$.\n\n\\medskip\n\nRecall that for any smooth compact tropical variety $\\mathscr W$, and for each integer $k$, we define the $k$-th cohomology of $\\mathscr W$ by\n\\[H^k(\\mathscr W) = \\sum_{p+q=k} H_\\trop^{p,q}(\\mathscr W).\\]\n\n\\medskip\n\nFor the inclusion $\\i\\colon\\mathcal Z\\hookrightarrow \\mathscr W$ of smooth compact tropical varieties, as in the local case, we get the corresponding restriction maps\n\\[\\i^*\\colon H^{k}(\\mathscr W) \\to H^{k}(\\mathcal Z),  \\]\nand the Gysin map\n\\[\\gys\\colon H^{k}(\\mathcal Z) \\to H^{k+2\\dim(\\mathscr W) - 2\\dim(\\mathcal Z)}(\\mathscr W),\\]\nwhich is the Poincar\\'e dual to the restriction map $H^{2\\dim(\\mathcal Z) - k}(\\mathscr W) \\to H^{2\\dim(\\mathcal Z)-k}(\\mathcal Z)$. The restriction maps respect the bi-degree $(p,q)$, while the Gysin map sends elements of bidegree $(p,q)$ to elements of bidegree $(p+\\dim(\\mathscr W)-\\dim(\\mathcal Z), q+\\dim(\\mathscr W)-\\dim(\\mathcal Z))$.\n\n\\medskip\n\n\\begin{prop}\\label{prop:adjunction} Let $\\mathcal Z$ be a divisor in $\\mathscr W$, and denote by $\\class(\\mathcal Z)$ the class of $\\mathcal Z$ in $H^{1,1}(\\mathscr W)$. For any $\\beta \\in H^k(\\mathscr W)$, we have\n\\[\\gys (\\i^*(\\beta)) = \\class(\\mathcal Z) \\cup \\beta.\\]\n\\end{prop}\n\\begin{proof} This follows for example by duality from the projection formula for tropical homology groups established in \\cite{GS-sheaf}*{Proposition 4.18}.\n\\end{proof}\n\nFrom the map of fans $\\pi\\colon  \\Delta' \\to \\Delta $, we get a projection map $\\pi \\colon\\TP_{\\Delta'} \\to \\TP_{\\Delta}$ which by restriction leads to a projection map\n$\\pi \\colon \\mathfrak X' \\to \\mathfrak X$. This map is \\emph{birational} in the sense that it induces an isomorphism on the open set $\\mathscr Y$ of $\\mathfrak X$ and $\\mathfrak X'$: in fact, the map $\\pi$ restricts to an isomorphism from $\\mathfrak X' \\setminus D'^\\rho $ to $\\mathfrak X \\setminus D^\\sigma$. Moreover, it sends $D'^\\rho$ to $D^\\sigma$.\n\nThe compositions of $\\gys$ and $\\pi^*$ gives now a map\n\\[\\gys \\circ\\,\\pi^*\\colon H^s(D^\\sigma) \\to H^{s+2}(\\mathfrak X').\\]\n\nWith these preliminaries, we can state the main theorem of this section.\n\\begin{thm}[Projective bundle formula] \\label{thm:keelglobal}\nWe have an isomorphism\n\\[H^k(\\mathfrak X' ) \\simeq H^k(\\mathfrak X) \\oplus T H^{k-2}(D^\\sigma) \\oplus T^2 H^{k-4}(D^\\sigma) \\oplus \\dots \\oplus T^{\\dims{\\sigma}-1}H^{k-2\\dims{\\sigma}+2}(D^\\sigma).\\]\n\\end{thm}\nHere the map from the right hand side to the left hand side restricts to $\\pi^*$ on $H^k(\\mathfrak X)$ and sends $T$ to $-\\class(D'^\\rho)$, so in view of Proposition~\\ref{prop:adjunction}, it is given on each factor $T^s H^{k-2s}(D^\\sigma)$ by\n\\[ \\begin{array}{rcl}\nT^s H^{k-2s}(D^\\sigma) & \\longrightarrow & H^k(\\mathfrak X') \\\\[1em]\nT^s \\alpha & \\longmapsto & (-1)^s \\class(D'^\\rho)^{s-1} \\cup \\gys\\circ\\pi^*(\\alpha).\n\\end{array} \\]\nThe decomposition stated in the theorem gives a decomposition for each pair of non-negative integers $(p,q)$\n\\[H^{p,q}(\\mathfrak X') \\simeq H^{p,q}(\\mathfrak X) \\oplus T H^{p-1, q-1}(D^\\sigma) \\oplus T^2 H^{p-2, q-2}(D^\\sigma) \\oplus \\dots \\oplus T^{\\dims{\\sigma}-1}H^{p-\\dims\\sigma+1,q - \\dims\\sigma+1}(D^\\sigma).\\]\n\n\\begin{remark} In this case, we have\n\\[T^{\\dims{\\sigma}} + c_1(N_{D^\\sigma})T^{\\dims\\sigma-1} +c_2(N_{D^\\sigma})T^{\\dims\\sigma-2} +\\dots + c_{\\dims \\sigma}(N_{D^\\sigma}) =0 \\]\nwhere $c_i(N_{D^\\sigma})$ are the Chern classes of the normal bundle $N_{D^\\sigma}$ in $\\mathfrak X$.\n\\end{remark}\nThe proof of this theorem is given in the next section. We actually give two proofs, one based on the use of the materials developed in the previous sections, using triangulations and the Steenbrink spectral sequence, and the other one based on Mayer-Vietoris exact sequence for tropical cohomology groups.\n\nIt is worth noting  that the second proof is more classical in spirit, while the first one being more interesting, requires more refined results about the geometry of tropical varieties which are not developed here and will only appear in our future work.\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:keelglobal} using triangulations}\n\nIn this section, we give a proof of Theorem~\\ref{thm:keelglobal} using the tropical Steenbrink spectral sequence and the isomorphism theorem (Theorem~\\ref{thm:steenbrink}). We need the following theorem.\n\n\\begin{thm} Suppose $\\mathscr Y$ is a polyhedral space in $\\mathbb{R}^n$, and let $\\Delta$ be a unimodular fan structure on $\\mathscr Y_\\infty$. There is a unimodular triangulation $Y$ of $\\mathscr Y$ with $Y_\\infty=\\Delta$.\n\\end{thm}\n\\begin{proof} In the case when $\\mathscr Y$ comes with a polyhedral structure $Y_0$ with recession fan $\\Delta$, this is proved in Section~\\ref{sec:triangulation}, \\cf. Theorem~\\ref{thm:unimodular_preserving_recession}. The point is to show that any polyhedral space $\\mathscr Y$ admits a polyhedral structure compatible with the choice of such a unimodular fan structure $\\Delta$, i.e., with recession fan $\\Delta$. This is more subtle and will be treated in our forthcoming work~\\cite{AP-geom}.\n\\end{proof}\n\nLet $Y$ be such a triangulation of $\\mathscr Y$. Let $X$ be its extensions to $\\mathfrak X$.\n\n\\begin{prop} The star subdivision $\\Delta'$ of $\\Delta$ induces a polyhedral structure $Y'$ on $\\mathscr Y$.\nMoreover, this polyhedral structure is unimodular and induces a unimodular polyhedral structure $X'$ on $\\mathfrak X'$ with the same finite part as $X$, i.e., $X_\\f = X'_\\f$.\n\\end{prop}\n\\begin{proof} The polyhedral structure $Y'$ is the one obtained as follows. Consider a face $\\eta$ in $Y$, and decompose $\\eta =\\eta_\\f+ \\eta_\\infty$, with $\\eta_\\f$ in $X_\\f$ and $\\eta_\\infty$ in the recession fan $Y_\\infty$. We define the subdivision $\\eta'$ of $\\eta$ obtained as the result of the barycentric star subdivision of $\\sigma$ in $\\eta_\\infty$: this means if $\\sigma$ is not a face of $\\eta_\\infty$, then $\\eta'=\\eta$ remains unchanged. On the other hand, if $\\sigma \\prec \\eta_\\infty$, then we subdivide $\\eta_\\f+ \\eta_\\infty$ by first taking the star subdivision $\\eta_\\infty'$ of $\\sigma$ in $\\eta_\\infty$, and define the subdivision $\\eta'$ as the one consisting of the faces of the form $\\gamma + \\tau$ with $\\gamma \\prec \\eta_\\f$ and $\\tau \\prec \\eta_\\infty'$.\n\\end{proof}\n\nThe following proposition describes the star fans of $Y'$ in terms of those of $Y$. By an abuse  of the notation, we also denote by $D^\\sigma$ the subdivision induced by $X$ on $D^\\sigma$.\n\n\\begin{prop} Let $\\eta \\in X_\\f =Y_\\f$ be a compact polyhedron. The star fan $\\Sigma'^{\\eta}$ of $\\eta$ in $X'$ is obtained from the star fan $\\Sigma^\\eta$ of $\\eta$ as the result of star subdividing $\\sigma$. In particular, if the star fan of $\\eta$ does not contain $\\sigma$, the two fans $\\Sigma'^{\\eta}$ and $\\Sigma^\\eta$ are isomorphic.\n\\end{prop}\nIn the above proposition, by star subdividing $\\sigma$ in $\\Sigma^\\eta$ we mean the fan obtained by the star subdivision of the image of $\\sigma$ in the space $N^\\eta_\\mathbb{R} = \\rquot{N_\\mathbb{R}} {N_{\\eta, \\mathbb{R}}}$, where we recall $N_{\\eta, \\mathbb{R}} =\\TT \\eta$.\n\\begin{proof} This follows directly from the description of the polyhedra in $Y'$.\n\\end{proof}\n\nIn the following, in order to distinguish the star fans and their cohomology in an ambient polyhedral complex, we use the following terminology. For a polyhedral complex $W$, and a face $\\eta$ of $W$, we write $\\Sigma^\\eta(W)$ in order to emphasize that the star fan $\\Sigma^\\eta$ is taken in $W$. We furthermore use $H^\\bul_W(\\eta)$ for the cohomology groups $H^\\bul\\bigl(\\,\\comp{\\Sigma^\\eta(W)}\\,\\bigr)$. In particular, $H^\\bul_{X}(\\eta)$ and $H^{\\bul}_{X'}(\\eta)$ denote the cohomology groups $H^\\bul\\bigl(\\,\\comp{\\Sigma^\\eta}\\,\\bigr)$ and $H^\\bul\\bigl(\\,\\comp{\\Sigma^{\\prime\\,\\eta}}\\,\\bigr)$, respectively.\n\n\\medskip\n\nBy Keel's lemma (Theorem~\\ref{thm:keel}) and the above proposition, for each face $\\eta \\in X_\\f$ whose star fan $\\Sigma^\\eta$ contains the cone $\\sigma$, we obtain a decomposition of the cohomology groups $H^k_{X'}(\\eta)$.\n\n\\begin{prop} For each cone $\\eta \\in X_\\f$ with $\\sigma \\in \\Sigma^\\eta$, we have\n\\[H^k_{X'}(\\eta) = H^k_X(\\eta) \\oplus T \\cdot H^{k-2}_{\\Sigma^\\eta}(\\sigma) \\oplus T^2 \\cdot H^{k-4}_{\\Sigma^\\eta}(\\sigma) \\oplus \\dots \\oplus T^{\\dims \\sigma-1} \\cdot H^{k- 2\\dims \\sigma+2}_{\\Sigma^\\eta}(\\sigma),\\]\nwhere $T$ is the class of $-x_\\rho \\in A^1(\\Sigma'^\\eta) = H^2_{X'}(\\eta)$.\n\\end{prop}\n\n\\medskip\n\nFor $W = X', X,$ or $D^\\sigma$, and for any integer $l$, we denote by $\\STp{l}_{W}^{\\bullet, \\bullet}$ the double complex we constructed in Section~\\ref{sec:steenbrinkdoublecomplex} whose total complex is the shifted cochain complex $\\ST_1^{\\bul, 2l}[l]$, the $2l$-th line of the Steenbrink spectral sequence calculated in $W$.\n\nLet $p$ be a non-negative integer. The previous proposition allows to get a decomposition of the double complex $\\STp{p}_{X'}^{\\bullet, \\bullet}$ in terms of the Steenbrink double complexes associated to $X$ and $D^\\sigma$.\n\n\\begin{prop} Notations as above, we have\n{ \\let\\oldoplus\\oplus \\renewcommand{\\oplus}{\\,\\,\\oldoplus\\,\\,}\n  \\let\\oldcdot\\cdot \\renewcommand{\\cdot}{\\!\\oldcdot\\!}\n\\[ \\STp{p}_{X'}^{\\bullet, \\bullet} = \\STp{p}_{X}^{\\bullet, \\bullet}  \\oplus T \\cdot \\STp{p-1}_{D^\\sigma}^{\\bullet, \\bullet-1} \\oplus T^2\\cdot\\STp{p-2}_{D^\\sigma}^{\\bullet, \\bullet-2}\\oplus{} \\cdots  \\oplus T^{\\dims\\sigma-1}\\cdot\\STp{p-\\dims{\\sigma}+1}_{D^\\sigma}^{\\bullet, \\bullet -\\dims{\\sigma}+1}. \\]\n}\n\\end{prop}\n\nThe map from the decomposition to $\\STp{p}_{X'}^{\\bullet, \\bullet}$ is made clear in the proof.\n\n\\begin{proof}\nBy definition of the Steenbrink double complexes, we have\n\\[ \\STp{p}^{a,b}_{X'}:=\\begin{cases}\n  \\displaystyle\\bigoplus_{\\eta\\in X'_\\f \\\\ \\dims\\eta=p+a-b} H^{2b}_{X'}(\\eta) & \\text{if $a\\geq0$ and $b\\leq p$}, \\\\[2em]\n  0 & \\text{otherwise,}\n\\end{cases} \\]\nwhose differential of bidegree $(1,0)$ is $\\i^*$ and whose differential of bidegree $(0,1)$ is $\\gys$. Note that we have $X_\\f = X'_\\f$. From the previous proposition, for any $\\eta \\in X_\\f$ whose star fan $\\Sigma^\\eta$ in $X$ contains $\\sigma$, we can decompose\n\\[H^{2b}_{X'}(\\eta) = H^{2b}_X(\\eta) \\oplus \\, T\\cdot \\,H^{2b-2}_{\\Sigma^\\eta}(\\sigma) \\oplus\\, T^2\\cdot H^{2b-4}_{\\Sigma^\\eta}(\\sigma) \\oplus \\dots \\oplus\\, T^{\\dims \\sigma-1}\\cdot H^{2b- 2\\dims \\sigma+2}_{\\Sigma^\\eta}(\\sigma). \\]\nHere $T$ is the class of $-x_\\rho$ in $H^2_{\\Sigma^\\eta}(\\sigma)$.\n\nMoreover, for any other polytope $\\eta\\in X_\\f$, we have $H^{2b}_{X'}(\\eta) = H^{2b}_{X}(\\eta)$.\n\n\\medskip\n\nLet $\\eta \\in X_\\f$ whose star fan $\\Sigma^\\eta$ contains $\\sigma$. Then the cone $\\sigma\\in\\Sigma^\\eta$ corresponds to the face $\\~\\eta:=\\comp{\\eta+\\sigma}\\in X$. We obtain a polytope $\\~\\eta\\cap D^\\sigma\\in X$ in $(D^\\sigma)_\\f$ for the polyhedral structure induced by $X$ on $D^\\sigma$. We denote this polytope by $\\eta_\\sigma$. The projection $\\pi_\\sigma\\colon \\mathscr Y\\to D^\\sigma$ along $\\TT\\sigma$ induces an isomorphism $\\eta\\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}}\\eta_\\sigma$. Moreover, the star fan of $\\sigma$ in $\\Sigma^\\eta$ is naturally isomorphic to the star fan of $\\~\\eta$ in $X$ and of $\\eta_\\sigma$ in $D^\\sigma$.\n\nUsing this observation, we rewrite the above decomposition as\n\\[H^{2b}_{X'}(\\eta) = H^{2b}_X(\\eta) \\oplus \\, T\\cdot \\,H^{2b-2}_{D^\\sigma}(\\eta_\\sigma) \\oplus\\, T^2\\cdot H^{2b-4}_{D^\\sigma}(\\eta_\\sigma) \\oplus \\dots \\oplus\\, T^{\\dims \\sigma-1}\\cdot H^{2b- 2\\dims \\sigma+2}_{D^\\sigma}(\\eta_\\sigma). \\]\n\nMoreover, the morphisms $\\i^*$ and $\\gys$ are compatible with this decomposition.\nIt follows that\n{ \\let\\oldoplus\\oplus \\renewcommand{\\oplus}{\\,\\,\\oldoplus\\,\\,}\n  \\let\\oldcdot\\cdot \\renewcommand{\\cdot}{\\!\\oldcdot\\!}\n\\[ \\STp{p}_{X'}^{\\bullet, \\bullet} = \\STp{p}_{X}^{\\bullet, \\bullet}  \\oplus T \\cdot \\STp{p-1}_{D^\\sigma}^{\\bullet, \\bullet-1} \\oplus T^2\\cdot\\STp{p-2}_{D^\\sigma}^{\\bullet, \\bullet-2}\\oplus{}\n\\cdots  \\oplus T^{\\dims\\sigma-1}\\cdot\\STp{p-\\dims{\\sigma}+1}_{D^\\sigma}^{\\bullet, \\bullet -\\dims{\\sigma}+1}. \\]\n}\nHere $T$ can be seen as the element of $\\STp{1}^{0,1}_{X'}\\subseteq\\ST_1^{0,2}(X')$ defined by\n\\[ T=\\sum_{v\\in X_\\f \\\\ \\dims{v}=1 \\\\ \\Sigma^v\\ni\\sigma}\\underbrace{-x_\\rho}_{\\in H^2_{X'}(v)}\\in\\bigoplus_{v\\in X_\\f \\\\ \\dims v=1}H^2_{X'}(v)\\subseteq\\STp{p}^{0,1}_{X'}. \\]\nThe multiplication by $T$ is similar to the multiplication by a K\u00e4hler class. It multiplies an element of $H^\\bul(\\eta)$ by $-x_\\rho\\in H^2(\\eta)$ if $\\sigma\\in\\Sigma^\\eta$, and by zero otherwise.\n\\end{proof}\n\n\\medskip\n\nWe are now ready to prove the projective bundle theorem.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:keelglobal}]\nApplying the previous proposition, and taking the total complexes on both sides, after shifting, gives a decomposition of cochain complexes\n\\[\\ST_1^{\\bul, 2p}(X') = \\ST_1^{\\bul, 2p}(X) \\oplus T\\cdot \\ST_1^{\\bul, 2p-2}(D^\\sigma)  \\oplus \\dots \\oplus T^{\\dims\\sigma-1}\\cdot \\ST_1^{\\bul, 2p-2\\dims\\sigma+2}(D^\\sigma).\\]\nTaking the cohomology of degree $q-p$ on both sides, and using our Isomorphism Theorem~\\ref{thm:steenbrink}, we get the decomposition\n\\[H^{p,q}(\\mathfrak X') \\simeq H^{p,q}(\\mathfrak X) \\oplus T H^{p-1, q-1}(D^\\sigma) \\oplus T^2 H^{p-2, q-2}(D^\\sigma) \\oplus \\dots \\oplus T^{\\dims{\\sigma}-1}H^{p-\\dims\\sigma+1,q - \\dims\\sigma+1}(D^\\sigma),\\]\nas stated in the theorem. Taking the sum over all $p,q$ with $p+q =k$, gives the decomposition for the $k$-th cohomology group of $X'$.\n\nIt remains to prove that $T$ corresponds to minus the cycle class associated to $D'^\\rho$ in $H^{1,1}(X')$. To see that, it suffices to compare the definition of $T$ in the proof of the previous proposition with the description of $\\class(D'^\\rho)$ in Section \\ref{sec:cl_D'}.\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:keelglobal} using Mayer-Vietoris}\n\nNotations as in the beginning of the section, let $\\mathfrak X$ and $\\mathfrak X'$ be the closure of $\\mathscr Y$ in $\\TP_\\Delta$ and $\\TP_{\\Delta'}$. We consider the open covering of $\\mathfrak X$ obtained by $\\mathscr V:=\\mathfrak X\\setminus D^\\sigma=\\mathfrak X'\\setminus D'^\\rho$ and a tubular neighborhood $\\mathscr U$ of $D^\\sigma$. Let $\\mathscr W$ be the intersection of $\\mathscr V$ and $\\mathscr U$. Let $\\pi\\colon \\mathfrak X'\\to \\mathfrak X$ be the projection map, and let $\\mathscr U' \\colon \\pi^{-1}(\\mathscr U)$, which is a tubular neighborhood of $D'^\\rho$. Note that $\\mathscr W = \\mathscr V \\cap \\mathscr U'$.\n\nThe Mayer-Vietoris exact sequences associated to these coverings of $\\mathfrak X'$ and $\\mathfrak X$ for tropical cohomology groups with coefficients in $\\SF^p$ lead to the following pair of long exact sequences where the vertical arrows are induced by the pullbacks.\n\n\\[\n\\begin{tikzcd}\n\\cdots \\rar& H^{p,q}(\\mathfrak X) \\dar \\rar & H^{p,q}(\\mathscr V)\\oplus H^{p,q}(\\mathscr U) \\dar \\rar & H^{p,q}(\\mathscr W) \\dar\\rar & H^{p,q+1}(\\mathfrak X) \\dar\\rar &  \\cdots \\\\\n\\cdots \\rar&  H^{p,q}(\\mathfrak X') \\rar & H^{p,q}(\\mathscr V)\\oplus H^{p,q}(\\mathscr U') \\rar & H^{p,q}(\\mathscr W) \\rar & H^{p,q+1}(\\mathfrak X') \\rar&  \\cdots\n\\end{tikzcd} \\]\n\nThe following is straightforward.\n\\begin{prop} The projection map from the tubular neighborhood to $D^\\sigma$ induces an isomorphism\n\\[H^{p,q} (\\mathscr U) \\simeq H^{p,q}(D^\\sigma)\\]\nfor any $q$. In a similar way, we get\n\\[H^{p,q} (\\mathscr U') \\simeq H^{p,q}(D'^\\rho).\\]\n\\end{prop}\n\n\\begin{prop}\\label{prop:proj_bundle_blowup} The projection map $D'^\\rho \\to D^\\sigma$ is a fibration with tropical projective spaces of dimension $\\dims\\sigma -1$.\n\\end{prop}\n\\begin{proof} Define the normal vector bundle $N_{D^\\sigma}$ associated to the embedding $D^\\sigma \\hookrightarrow \\mathfrak X$.\n\\[D^\\rho \\simeq \\TP(N_{D^\\sigma}), \\]\nthe projectivization of $N_{D^\\sigma}$, from which the result follows.\n\\end{proof}\n\n\\begin{thm}\\label{thm:proj_bundle_thm} Let $E$ be a vector bundle of rank $r$ over a compact tropical variety $\\mathcal Z$, and let $\\mathcal Z' = \\TP(E)$ be the projectivization of $E$. Then we get an isomorphism\n\\[H^{p,q}(\\mathcal Z') \\simeq H^{p,q}(\\mathcal Z) \\oplus T H^{p-1,q-1}(\\mathcal Z) \\oplus T^2 H^{p-2,q-2}(\\mathcal Z) \\oplus \\dots \\oplus T^{r-1}H^{p-r+1,q-r+1}(\\mathcal Z).\\]\n\\end{thm}\n\n\\begin{proof} Applying the Mayer-Vietoris exact sequence, we reduce to the case where the vector bundle is trivial, i.e., $E = \\mathbb{R}^r\\times \\mathcal Z$. In this case, we have $\\TP(E) = \\TP^{r-1} \\times \\mathcal Z$. The theorem now follows by K\\\"unneth decomposition theorem for tropical cohomology groups~\\cite{GS-sheaf}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:keelglobal}] The theorem follows by combining the two Mayer-Vietoris long exact sequences associated to covers $\\{\\mathscr V, \\mathscr U\\}$ and $\\{\\mathscr V, \\mathscr U'\\}$ of $\\mathfrak X$ and $\\mathfrak X'$, and Theorem~\\ref{thm:proj_bundle_thm} applied to the fibration $D'^\\rho \\to D^\\sigma$, by using Proposition~\\ref{prop:proj_bundle_blowup}, which allows to decompose the cohomology of $D'^\\rho$ in terms of the cohomology of $D^\\sigma$.\n\\end{proof}\n\n\n\n\n\n\n\\section{Proof of Theorem~\\ref{thm:main2}} \\label{sec:proofmaintheorem}\nIn this section, we prove Theorem~\\ref{thm:main2}. We start by recalling the statement of the theorem. Let $\\mathscr Y$ be a smooth rational polyhedral space in $V = N_\\mathbb{R}  \\simeq \\mathbb{R}^n$. Let $\\Delta$ be a unimodular recession fan on $\\mathscr Y_\\infty$. Let $\\mathfrak X$ be the closure of $\\mathscr Y$ in $\\TP_\\Delta$. Let $\\ell$ be a strictly convex function on $\\Delta$. Then, $\\ell$ defines an element of $H^{1,1}(\\mathfrak X)$ given by\n\\[ \\sum_{\\varrho\\in\\Delta \\\\ \\dims\\varrho=1}\\ell(\\varrho)\\class(D^\\varrho). \\]\nBy an abuse of the notation, we denote this element and the corresponding Lefschetz operator on $H^{\\bul, \\bul}(\\mathfrak X)$ by $\\ell$ as well.\n\nWorking with $\\mathbb{Q}$ coefficients, we have to prove the following statements.\n\\begin{itemize}\n\\item (Weight-monodromy conjecture) For $q>p$ two non-negative integers, we get an isomorphism\n\\[N^{q-p} \\colon H_{\\trop}^{q,p}(\\mathfrak X) \\to H_{\\trop}^{p,q}(\\mathfrak X).\\]\n\n\\item (Hard Lefschetz) For $p+q \\leq d\/2$, the Lefschetz operator $\\ell$ induces an isomorphism\n\\[\\ell^{d- p-q}\\colon H_{\\trop}^{p,q}(\\mathfrak X) \\to H^{d-q, d-p}_{\\trop}(\\mathfrak X).\\]\n\n\\item (Hodge-Riemann) The pairing $(-1)^p \\bigl< \\,\\cdot\\,,\\, \\ell^{d-p-q} N^{q-p}\\,\\cdot\\, \\bigr>$ induces a positive-definite pairing on the primitive part $P^{p,q}$ of $H_{\\trop}^{p,q}$, where $\\bigl< \\,\\cdot\\,,\\cdot\\, \\bigr>$ is the natural pairing\n\\[\\bigl< \\,\\cdot\\,, \\cdot\\,\\bigr> \\colon H^{q,p}_\\trop(\\mathfrak X) \\otimes H_\\trop^{d-q,d-p}(\\mathfrak X) \\to H_\\trop^{d,d}(\\mathfrak X) \\simeq \\mathbb{Q}.\\]\n\\end{itemize}\n\n\n\n\\subsection{Global ascent-descent} Situation as in the previous section, let $\\Delta = \\mathscr Y_\\infty$ be unimodular and let $\\sigma$ be a cone in $\\Delta$. Let $\\Delta' $ be the star subdivision of $\\Delta$ obtained by star subdividing $\\sigma$. Denote by $\\rho$ the new ray in $\\Delta'$. For an element $\\omega$ in $H^{1,1}(\\mathfrak X)$, we denote by $\\omega^\\sigma$ the restriction of $\\omega$ to $D^\\sigma$, i.e., the image of $\\omega$ under the restriction map $H^{1,1}(\\mathfrak X) \\to H^{1,1}(D^\\sigma)$, for the inclusion of tropical varieties $D^\\sigma \\hookrightarrow \\mathfrak X$.\n\nLet $\\ell$ be a strictly convex cone-wise linear function on $\\Delta$. Denote by $\\ell'$ the convex function induced by $\\ell$ on $\\Delta'$. As in the previous section, we denote by $\\ell$ and $\\ell'$ the corresponding elements of $H^{1,1}(\\mathfrak X)$ and $H^{1,1}(\\mathfrak X')$, respectively.\n\nThe strictly convex function $\\ell$ induces a strictly convex function $\\ell^\\sigma$ on the star fan $\\Sigma^\\sigma$ of $\\sigma$ in $\\Delta$. By identification of $\\Sigma^\\sigma$ with the recession fan of $D^\\sigma$, we get a strictly convex function $\\ell^\\sigma$ on the recession fan $D^\\sigma_\\infty$. This defines an element in $H^{1,1}(D^\\sigma)$ that we denote with our convention above by $\\ell^\\sigma$.\n\n\\begin{prop} \\label{prop:class_commutes}\nThe element $\\ell^\\sigma \\in H^{1,1}(D^\\sigma)$ coincides with the restriction of $\\ell \\in H^{1,1}(X)$ to $D^\\sigma \\hookrightarrow \\mathfrak X$.\n\\end{prop}\n\\begin{proof}\nWe can assume without loss of generality that $\\ell$ is zero on $\\sigma$ and that $\\ell^\\sigma$ is the pushforward of $\\ell$ by the projection $\\Sigma \\to \\Sigma^\\sigma$. Denote by $D_\\ell$ the Poincar\u00e9 dual of $\\ell$ in $H^\\trop_{1,1}(\\mathfrak X)$, and by $D_{\\ell^\\sigma}$ the Poincar\u00e9 dual of $\\ell^\\sigma$ in $H^\\trop_{1,1}(D_\\sigma)$. By Poincar\u00e9 duality, it suffices to prove that the cap product $D_\\ell \\cap D^\\sigma\\in H^\\trop_{1,1}(D^\\sigma)$ equals $D_{\\ell^\\sigma}$. For a ray $\\varrho$ of $\\Sigma$, we write $\\varrho\\sim\\sigma$ if $\\varrho$ is in the link of $\\sigma$, i.e., if $\\sigma+\\varrho$ is a face of $\\Sigma$ and $\\varrho\\not\\in\\sigma$. From our assumption that $\\ell$ is zero on $\\sigma$, we get\n\\[ D_\\ell \\cap D^\\sigma = \\sum_{\\varrho\\in\\Sigma \\\\ \\dims\\varrho=1}\\ell(\\varrho)(D^\\varrho\\cap D^\\sigma) = \\sum_{\\varrho\\sim\\sigma \\\\ \\dims\\varrho=1}\\ell(\\varrho)(D^\\varrho\\cap D^\\sigma) = D_{\\ell^\\sigma}. \\qedhere \\]\n\\end{proof}\n\nWe now state and prove the following global version of Theorem~\\ref{thm:barycentric_subdivision}.\n\n\\begin{thm} \\label{thm:global_ascent_descent} Notations as above, we have the following properties.\n\\begin{itemize}\n\\item \\emph{(Ascent)} Assume the property $\\HR(D^\\sigma,\\ell^\\sigma)$ holds. Then $\\HR(\\mathfrak X, \\ell)$ implies $\\HR\\bigl(\\mathfrak X',\\ell'-\\epsilon\\class(D^\\rho)\\bigr)$ for any small enough $\\epsilon>0$.\n\n\\item \\emph{(Descent)} We have the following partial inverse: if both the properties $\\HR(D^\\sigma,\\ell^\\sigma)$ and $\\HL(\\mathfrak X,\\ell)$ hold, and if we have the property $\\HR\\bigl(\\mathfrak X',\\ell'-\\epsilon\\class(D^\\rho)\\bigr)$ for any small enough $\\epsilon>0$, then we have $\\HR(\\mathfrak X, \\ell)$.\n\\end{itemize}\n\\end{thm}\n\n\\begin{proof}\nThe projective bundle formula (Theorem \\ref{thm:proj_bundle_thm}) is the global analog of local Keel's lemma (Theorem \\ref{thm:keel}). Moreover Poincar\u00e9 duality holds for $\\mathfrak X$ and $\\mathfrak X'$. Thus, by Remark \\ref{rem:keel}, the proof is identical to that of the corresponding local theorem, Theorem~\\ref{thm:barycentric_subdivision}.\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:main2}} We proceed as in the proof of the properties $\\HR$ and $\\HL$ in the local case.\n\nSo let $\\mathscr Y$ be a smooth tropical variety of dimension $d$ in $V \\simeq \\mathbb{R}^n$. Any unimodular recession fan $\\Delta$ on $\\mathscr Y_\\infty$ provides a compactification of $\\mathscr Y$ by taking its closure in $\\TP_\\Delta$. We denote by $\\mathfrak X_\\Delta$ this compactification in order to emphasize the dependence in $\\Delta$. Since $\\mathscr Y$ is smooth, the variety $\\mathfrak X_\\Delta$ for any unimodular fan $\\Delta$ is a smooth tropical variety.\n\n\\medskip\n\nFrom now on we restrict to fans $\\Delta$ on the support of $\\mathscr Y_\\infty$ which are in addition quasi-projective.\nWe have to show that for any such fan $\\Delta$ and any strictly convex function $\\ell$ on $\\Delta$, the pair $(H^\\bul(X), \\ell)$ verifies $\\HR$, $\\HL$, and the weight-monodromy conjecture.\n\n\\medskip\n\nProceeding by induction, we suppose that the theorem holds for all smooth projective tropical varieties of dimension strictly smaller than $d$.\n\n\\medskip\n\nAs in the local case, the following proposition is the base of our proof. It allows to start with a fan structure on the asymptotic cone of $\\mathscr Y$ such that the corresponding compactification of $\\mathscr Y$ verifies the Hodge-Riemann relations, and then use the global ascent and descent properties to propagate the property to any other smooth compactification of $\\mathscr Y$, given by any other fan structure on $\\mathscr Y_\\infty$.\n\n\\begin{prop}\\label{prop:baseHR-global}\nNotations as above, there exists a quasi-projective unimodular fan $\\Delta_0$ on $\\mathscr Y_\\infty$ such that, for any strictly convex element $\\ell_0$ on $\\Delta_0$, we have the properties $\\HR(\\mathfrak X_{\\Delta_0}, \\ell_0)$ and $\\HL(\\mathfrak X_{\\Delta_0}, \\ell_0)$ for the compactification $\\mathfrak X_{\\Delta_0}$ of $\\mathscr Y$ in $\\TP_{\\Delta_0}$.\n\\end{prop}\n\\begin{proof} By Theorem~\\ref{thm:triangulation_unimodulaire_convexe}, replacing the lattice $N$ with $\\frac 1k N$ for some integer $k$, we can find a projective unimodular triangulation $Y_0$ of $\\mathscr Y$ such that the recession fan $\\Delta_0$ of $Y_0$ is quasi-projective and unimodular.\n\nLet $X_{\\Delta_0}$ be the compactification of $Y_0$ in $\\TP_{\\Delta_0}$. Let $\\~\\ell_0$ be a strictly convex function on $Y_0$. By the second part of Theorem~\\ref{thm:triangulation_unimodulaire_convexe}, we can assume that $\\ell_0:=(\\~\\ell_0)_\\infty$ is a well-defined and strictly convex function on $\\Delta_0$. By theorem \\ref{thm:projective_Kahler}, $\\~\\ell_0$ defines a K\u00e4hler class of $X_{\\Delta_0}$. Moreover, by Corollary \\ref{cor:class_independent_finite_part} and Remark \\ref{rem:explicit_function}, this class equals $\\class(\\ell_0)\\in H^{1,1}(\\mathfrak X_{\\Delta_0})$. It thus follows from our Theorem~\\ref{thm:main} that the pair $(X_{\\Delta_0}, \\ell_{0})$ verifies $\\HL$ and $\\HR$.\n\\end{proof}\n\n\\begin{prop} \\label{prop:HR-oneall-global}\nFor any unimodular fan $\\Delta$ on the support of $Y_\\infty$, we have the equivalence of the following statements.\n\\begin{itemize}\n\\item $\\HR(X_\\Delta, \\ell)$ is true for any strictly convex function $\\ell$ on $\\Delta$.\n\\item $\\HR(X_\\Delta, \\ell)$ is true for one strictly convex element $\\ell$ on $\\Delta$.\n\\end{itemize}\n\\end{prop}\nThe proof uses the following proposition.\n\\begin{prop} \\label{prop:global_HR}\nIf $\\HR(D^\\varrho, \\ell^\\varrho)$ holds for all rays $\\varrho\\in\\Delta$, then we have $\\HL(X, \\ell)$.\n\\end{prop}\n\\begin{proof}\nBy Proposition \\ref{prop:class_commutes}, we know that $\\ell^\\varrho=\\i^*(\\ell)$ where $\\i$ is the inclusion $D^\\varrho\\hookrightarrow X$. One can now adapt the proof of \\ref{prop:local_HR} to the global case to prove the proposition.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:HR-oneall-global}] Let $\\ell$ be a strictly convex piecewise linear function on $\\Delta$. For all $\\varrho \\in \\Delta$, we get a strictly convex function $\\ell^\\varrho$ on the star fan $\\Sigma^\\varrho$ of $\\varrho$ in $\\Delta$. By the hypothesis of our induction, we know that $\\HR(D^\\varrho, \\ell^\\varrho)$ holds. By Proposition~\\ref{prop:global_HR} we thus get $\\HL(X_\\Delta, \\ell)$.\n\nBy Proposition \\ref{prop:HRbis}, we get that $\\HR(X_\\Delta,\\ell)$ is an open and closed condition on the set of all $\\ell$ which satisfy $\\HL(X_\\Delta,\\ell)$. In particular, if there exists $\\ell_0$ in the cone of strictly convex elements which verifies $\\HR(X_\\Delta,\\ell_0)$, any $\\ell$ in this cone should verify $\\HR(X_{\\Delta},\\ell)$.\n\\end{proof}\n\n\\medskip\n\nLet $\\Delta$ be a unimodular fan of dimension $d$ on $\\mathscr Y_\\infty$, let $\\ell$ be a convex piecewise linear function on $\\Delta$, and let $\\Delta'$ be the fan obtained from $\\Delta$ by star subdividing a cone $\\sigma\\in\\Delta$. Denote by $\\rho$ the new ray in $\\Delta'$, and let $\\ell'$ be the piecewise linear function induced by $\\ell$ on $\\Delta'$.\n\nDenote by $\\chi_\\rho$ the characteristic function of $\\rho$ on $\\Delta'$ which takes value one on $\\e_\\rho$, value zero on each $\\e_\\varrho$ for all ray $\\varrho \\neq \\rho$ of $\\Delta'$, and which is linear on each cone of $\\Delta'$.\nThe following is straightforward.\n\n\\begin{prop} For any small enough $\\epsilon>0$, the function $\\ell' - \\epsilon \\chi_\\rho$ is strictly convex. \\end{prop}\n\n\\begin{prop} \\label{prop:HR-trans-global}\nNotations as above, the following statements are equivalent.\n\\begin{enumerate}[label={\\textnormal{(\\textit{\\roman*})}}]\n\\item \\label{enum:HR-trans-global:i} We have $\\HR(\\mathfrak X_\\Delta, \\ell)$.\n\\item \\label{enum:HR-trans-global:ii} The property $\\HR(\\mathfrak X_{\\Delta'},\\ell'-\\epsilon \\chi_\\rho)$ holds for any small enough $\\epsilon>0$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nThis follows from Theorem~\\ref{thm:global_ascent_descent}. We first observe that the $(1,1)$-class associated to the convex function $\\ell'-\\epsilon \\chi_\\rho$ is precisely $\\ell' - \\epsilon \\class(D^\\rho)$, where in this second expression $\\ell'$ represents the class of the convex function $\\ell'$ in $H^{1,1}(X_{\\Delta'})$. Moreover, by the hypothesis of our induction, we have $\\HR(D^\\sigma, l_\\sigma)$. Now, a direct application of the ascent part of Theorem~\\ref{thm:global_ascent_descent} leads to the implication $\\ref{enum:HR-trans-global:i} \\Rightarrow \\ref{enum:HR-trans-global:ii}$.\n\nWe now prove $\\ref{enum:HR-trans-global:ii} \\Rightarrow \\ref{enum:HR-trans-global:i}$. Proposition \\ref{prop:global_HR} and our induction hypothesis imply that we have $\\HL(X_{\\Delta}, \\ell)$. Applying now the descent part of Theorem~\\ref{thm:global_ascent_descent} gives the result.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main2}] By Proposition~\\ref{prop:baseHR-global}, there exists a fan $\\Delta_0$ with support $\\mathscr Y_\\infty$ and a strictly convex element $\\ell_0$ on $\\Delta_0$ such that the result holds for the pair $(\\mathfrak X_{\\Delta_0}, \\ell_0)$. By Propositions~\\ref{prop:HR-trans-global} and~\\ref{prop:HR-oneall-global}, for any quasi-projective unimodular fan $\\Delta$ with support $\\mathscr Y_\\infty$ which can be obtained from $\\Delta_0$ by a sequence of star subdivisions and star assemblies, and for any strictly convex function $\\ell$ on $\\Delta$, we get $\\HR(\\mathfrak X_{\\Delta}, \\ell)$. The properties $\\HR$ and $\\HL$ both follow by applying Theorem~\\ref{thm:equivalent_fan2}.\n\n\\medskip\n\nThe weight-monodromy conjecture is preserved by the projective bundle formula, Theorem~\\ref{thm:keelglobal}. Thus the property propagates from $\\Delta_0$ to all quasi-projective fans by Theorem~\\ref{thm:equivalent_fan2}.\n\\end{proof}\n\n\\newpage\n\n\n\n\n\n\n\\section*{Appendix}\n\n\\section{Spectral resolution of the tropical complex}\n\\label{sec:technicalities}\n\n{\\renewcommand{\\\/}{\\backslash}\n\\allowdisplaybreaks\n\\def\\sign(#1,#2){\\varepsilon_{#2\/#1}}\n\nIn this section, we prove that the objects used to apply the spectral resolution lemma in Section~\\ref{sec:steenbrink} verify the hypothesis of this lemma. We also prove Proposition~\\ref{prop:isomorphism_filtrations}, which states that the isomorphism between the tropical complex $C^{p,\\bul}$ and $\\Tot^\\bul(\\CCp{p}^{\\bul,\\bul})$ is an isomorphism of filtered complexes for the corresponding differentials and filtrations. We recall all the objects below.\n\n\\medskip\n\nWe suppose $\\mathfrak X$ is a compact smooth tropical variety with a unimodular triangulation that we denote $X$. We suppose in addition that the underlying polyhedral space $\\mathfrak X$ is obtained by compactifying a polyhedral space $\\mathscr Y$ in $\\mathbb{R}^n$. This will allow later to fix a scalar product in $\\mathbb{R}^n$ and define \\emph{compatible} projections on subspaces of $\\mathbb{R}^n$, in a sense which will be made clear.\n\n\n\n\\subsection{The triple complex $\\AA^{\\bul,\\bul,\\bul}$} We start by recalling the definition of the triple complex $\\AA^{\\bul, \\bul, \\bul}$, which is the main object of study in this section (and which is roughly obtained from $\\Da{\\bul}^{\\bul,\\bul}$ by inserting the Steenbrink and tropical chain complexes into it).\n\n\\medskip\n\nLet $p$ be a non-negative integer. The triple complex $\\AA^{\\bul, \\bul, \\bul}$ is defined as follows. We set\n\\[\\AA^{\\bul, \\sqbullet, -1} := \\CCp{p}^{\\bul,\\sqbullet}, \\qquad \\AA^{\\bul,-1,\\blackdiamond} := \\STinf{p}^{\\bul,\\blackdiamond}, \\, \\textrm{and} \\qquad \\AA^{\\bul,\\sqbullet,\\blackdiamond} := \\Da{\\bul}^{\\sqbullet,\\blackdiamond}.\\]\nMore explicitly, let $a,b,b'\\geq 0$ be three non-negative integers. We set\n\n\\medskip\n\n\\begin{itemize}[label=-]\n\\item \\( \\displaystyle \\AA^{a,b,-1} := \\bigoplus_{\\dims\\delta = a+b} \\AA^{a,b,-1}[\\delta] \\quad \\text{ with } \\quad   \\AA^{a,b,-1}[\\delta]  = \\bigwedge^b\\TT^\\dual\\delta \\otimes \\SF^{p-b}(\\conezero^\\delta). \\)\n\n\\medskip\n\n\\item \\( \\displaystyle \\AA^{a,-1,b'} = \\begin{cases}\n  \\bigoplus_{\\dims\\eta = p+a-b' \\\\ \\dims{\\eta_\\infty} \\geq a} \\AA^{a,-1, b'}[\\eta] & \\text{if $b'\\geq$ p,} \\\\\n  0 & \\text{otherwise,}\n\\end{cases} \\quad \\text{ with } \\quad \\AA^{a,-1,b'}[\\eta] := H^{2b'}(\\eta).\n\\)\n\n\\medskip\n\n\\item \\( \\AA^{a,b,b'} = \\Dnop^{a,b,b'} = \\bigoplus_{\\delta\\prec\\eta \\\\ \\sed{\\delta}=\\sed{\\eta} \\\\ \\dims{\\delta}=a+b \\\\ \\dims{\\eta}=p+a-b' }\\AA^{a,b,b'}[\\delta,\\eta] \\quad \\text{ with }\\quad \\AA^{a,b,b'}[\\delta,\\eta] := \\bigwedge^b\\TT^\\dual\\delta \\otimes H^{2b'}(\\eta).\\)\n\n\\medskip\n\n\\item All the other pieces $\\AA^{a,b,b'}$ are trivial.\n\\end{itemize}\n\n\\medskip\n\nWe now set $\\AA_\\trop^{\\bul,\\sqbullet}:=\\AA^{\\bul,\\sqbullet,-1}$, $\\AA_\\ST^{\\bul,\\blackdiamond}:=\\AA^{\\bul,-1, \\blackdiamond}$, and define $\\AA_\\Dnop^{\\bul,\\bul,\\bul}$ as the truncation of $\\AA^{\\bul,\\bul, \\bul}$ at non-negative indices, i.e., such that\n\\[ \\AA_\\Dnop^{a,b,b'}=\\begin{cases}\n\\AA^{a,b,b'} & \\text{if $a,b,b'\\geq0$}, \\\\\n0 & \\text{otherwise.}\n\\end{cases} \\]\nNote that $\\AA_\\Dnop^{a,b,b'} = \\Da{a}^{b,b'}$ for the double complex $\\Da{a}^{b,b'}$ of Section~\\ref{sec:steenbrink}.\n\n\\medskip\n\nFor $\\sigma\\in X_\\infty$, we set\n\\[ \\AAa{\\sigma}^{\\bul,\\bul,\\bul}:=\\bigoplus_{\\delta \\text{ with }  \\sed(\\delta) = \\sigma} \\AA^{\\bul,\\bul,\\bul}[\\delta], \\]\nand define $\\AAa{\\sigma}^{\\bul, \\bul}_\\trop, \\AAa{\\sigma}^{\\bul, \\bul}_\\ST$, and $\\AAa{\\sigma}^{\\bul, \\bul, \\bul}_\\Dnop$, similarly.\n\nThe differentials on $\\AA^{a,b,b'}$ of respective degree $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ are denoted $\\dfrak$, $\\d$ and $\\d'$.\nWe will recall all these maps in Section~\\ref{sec:maps}. Their restrictions to $\\AA^{\\bul,\\bul}_\\trop$ are denoted by $\\dfrak_\\trop, \\d_\\trop$ and $\\d'_\\trop$ respectively. We only restrict the domain here, and not the codomain. Thus $\\d'_\\trop$ is a morphism from $\\AA^{\\bul,\\bul}_\\trop$ to $\\AA^{\\bul,\\bul,\\bul}$. We use the same terminology for the two other complexes $\\AA^{\\bul,\\bul}_\\ST$, and $\\AA_D^{\\bul,\\bul,\\bul}$.\n\n\\medskip\n\nThe notation $\\d\\bigl[\\AA^{a,b,b'}[\\eta] \\to \\AA^{a,b+1,b'}[\\mu]\\bigr]$ denotes restriction of the differential $\\d$ to $\\AA^{a,b,b'}[\\eta]$ projected to the piece $\\AA^{a,b+1,b'}[\\mu]$, i.e., this is the composition $\\pi\\d\\i$ where $\\i\\colon\\AA^{a,b,b'}[\\eta]\\hookrightarrow\\AA^{\\bul,\\bul,\\bul}$ is the natural inclusion and $\\pi\\colon\\AA^{\\bul,\\bul,\\bul}\\twoheadrightarrow\\AA^{a,b+1,b'}[\\mu]$ is the natural projection.\n\nMoreover, the star $*$ in the expressions $\\d[* \\to \\AA^{a,b+1,b'}]$ and $\\d[\\AA^{a,b,b'} \\to *]$, denotes the natural domain and codomain of $\\d$, i.e., $\\AA^{\\bul,\\bul,\\bul}$, respectively. Similarly, in the expressions $\\d[\\, {\\prescript{}{\\sigma}{*}} \\to \\AA^{a,b+1,b'}]$, $\\d[\\AA^{a,b,b'} \\to {\\prescript{}{\\sigma}{*}}\\,]$, and $\\d[\\,{\\prescript{}{\\sigma}{*}} \\to {\\prescript{}{\\sigma}{*}}\\,]$, the ${\\prescript{}{\\sigma}{*}}$ denotes $\\AAa{\\sigma}^{\\bul, \\bul,\\bul}$.\n\nWe use similar notations for other differentials and other pieces of $\\AA^{\\bul,\\bul,\\bul}$.\n\n\\medskip\n\nWe decompose the differential $\\dfrak$ as the sum $\\d^\\i+\\d^\\pi$ where\n\\begin{gather*}\n\\d^\\i := \\bigoplus_{\\sigma\\in X_\\infty}\\dfrak[\\,{\\prescript{}{\\sigma}{*}} \\to {\\prescript{}{\\sigma}{*}}\\,], \\text{ and } \\\\\n\\d^\\pi := \\bigoplus_{\\tau,\\sigma\\in X_\\infty \\\\ \\sigma\\ssupface\\tau}\\dfrak[\\,{\\prescript{}{\\sigma}{*}} \\to {\\prescript{}{\\tau}{*}}\\,].\n\\end{gather*}\n\nThe two other differential $\\d$ and $\\d'$ preserve $\\AAa{\\sigma}^{\\bul,\\bul,\\bul}$ for any $\\sigma\\in X_\\infty$.\n\n\\medskip\n\nThe first main result of this section is the following.\n\\begin{prop} \\label{prop:tech_1}\nThe complex $\\Tot^\\bul(\\AA_\\trop^{\\bul,\\bul}, \\dfrak_\\trop+\\d_\\trop)$ is isomorphic as a complex to $(C_\\trop^{p,\\bul}, \\partial_\\trop)$ where $\\partial_\\trop$ is the usual tropical differential.\n\\end{prop}\n\nThe second result is Proposition~\\ref{prop:differential_triple_D}, which, we recall, states the following.\n\\begin{prop} \\label{prop:tech_2}\nThe map $\\partial=\\dfrak+\\d+\\d'$ is a differential, i.e.,\n\\[ \\partial\\partial = 0. \\]\n\\end{prop}\n\nThis section is supposed to be self-contained, in particular we will recall the maps below. We start by (re)introducing some notations and making some conventions. We note that a few of the notations below differ from the ones used in the previous sections.\n\n\n\n\\subsection{Notations and conventions}\nHere is a set of conventions and notations we will use.\n\n\n\\subsubsection*{Naming convention on faces}\nIn what follows, we take faces $\\gamma\\ssubface\\,\\delta,\\!\\delta'\\,\\ssubface\\chi$ forming a diamond and $\\zeta\\ssubface\\,\\eta,\\!\\eta'\\,\\ssubface\\mu$ forming another diamond such that $\\gamma \\prec \\zeta$, $\\delta \\prec \\eta$, $\\delta' \\prec \\eta',$ and $\\chi \\prec \\mu$.\n\nMoreover, we make the convention that if a relation involves both $\\gamma$ and $\\delta$, then it is assumed that $\\gamma\\ssubface\\delta$. However, for this relation, we do \\emph{not a priori assume} that there exists a face $\\chi$ such that $\\chi \\ssupface\\delta$: $\\delta$ might be a face of maximum dimension. Unless stated otherwise, we assume that all these faces have the same sedentarity.\n\nMoreover, $\\tau\\ssubface\\sigma\\prec\\xi$ will be all the time three cones of $X_\\infty$.\n\n\\medskip\n\n\n\\subsubsection*{Convention on maps $\\pi, \\iota, \\i$ and $\\p$}\nIf $\\phi$ and $\\psi$ are two faces of $X$ or cones of $\\comp{X_\\infty}$, we define, when this has a meaning, the map\n$\\pi^{\\phi\/\\psi}\\colon N^\\psi_\\mathbb{R} \\twoheadrightarrow N^\\phi_\\mathbb{R}$ to be the natural projection. Recall that for $\\phi$ a face of $X$ of sedentarity $\\conezero$ or a cone of $X_\\infty$, we define $N_{\\phi,\\mathbb{R}}=\\TT\\phi$ and $N^\\phi_\\mathbb{R} = \\rquot{N_\\mathbb{R}}{N_{\\phi,\\mathbb{R}}}$, and extend this to faces and cones of higher sedentarity by $N^\\phi_\\mathbb{R} = \\rquot{N^{\\sed(\\phi)}_{\\mathbb{R}}}{\\TT\\phi}$.\n\n\\medskip\n\nFor the map $\\pi^{\\phi\/\\psi}$, it might happen that we use the notation $\\pi^{\\psi\\\/\\phi}$ in order to insist on the fact that the domain of the map concerns the face $\\psi$ and the codomain $\\phi$.\n\n\\medskip\n\nTo simplify the notations, if $\\psi=\\sed(\\phi)$, we set $\\pi^\\phi:=\\pi^{\\psi\\\/\\phi}$.\nSometimes, we will put the superscript as a subscript instead, for example $\\pi_{\\phi\/\\psi}^*$ is another notation for $\\pi^{\\phi\/\\psi *}$. Moreover, depending on the context and when this is clear, we use $\\pi^{\\psi\\\/\\phi}$ to denote as well the natural projection $\\bigwedge^k N^\\phi_\\mathbb{R} \\twoheadrightarrow \\bigwedge^k N^\\psi_\\mathbb{R}$, and also the restriction of this projection to some subspace.\n\nTo give an example of these conventions, for instance, the map $\\SF^k(\\delta) \\twoheadrightarrow \\SF^k(0^\\delta)$ is represented by the notation $\\pi^\\delta$.\n\n\\medskip\n\nWe denote by $\\eta^\\delta$ the face corresponding to $\\eta$ in $\\Sigma^\\delta$, i.e., $\\eta^\\delta=\\mathbb{R}_+\\pi^\\delta(\\eta)$. In the same way, if $\\sigma \\prec \\delta_\\infty$, we set\n\\[\\delta_\\infty^\\sigma:=\\pi^\\sigma(\\delta)=\\rquot{\\bigl(\\delta + N_{\\sigma,\\mathbb{R}}\\bigr)}{N_{\\sigma,\\mathbb{R}}}.\n\\]\nIf $\\sed(\\delta)=\\conezero$, it is the intersection of the closure of $\\delta$ in $\\TP_{X_\\sminfty}$ with the stratum of sedentarity $\\sigma$.\n\n\\medskip\n\nIf $\\gamma\\ssubface\\delta$ are two faces of different sedentarities, then $\\pi^{\\sed(\\delta)\\\/\\sed(\\gamma)}$ induces a projection from $\\TT\\delta$ to $\\TT\\gamma$. We denote this projection by $\\pi^{\\sed}_{\\delta\/\\gamma}$. Note that this is quite different from $\\pi^{\\gamma\\\/\\delta}$; for such a pair, the projection $\\pi^{\\gamma\\\/\\delta}$ is in fact an isomorphism $N^\\delta_\\mathbb{R} \\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} N^\\gamma_\\mathbb{R}$ and induces an isomorphism $\\Sigma^\\delta \\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} \\Sigma^\\gamma$. We denote these maps and the corresponding pushforwards by $\\id^{\\delta\/\\gamma}$, and the inverse maps and the corresponding pushforwards by $\\id^{\\gamma\\\/\\delta}$.\n\n\\medskip\n\nIn the same way, we set $\\iota_{\\eta\\\/\\delta}\\colon \\SF_\\bul\\eta \\hookrightarrow \\SF_\\bul\\delta$ and $\\iota_{\\delta}\\colon \\SF_\\bul\\delta \\hookrightarrow \\bigwedge^\\bul N^{\\sed(\\delta)}_\\mathbb{R}$ to be the natural inclusions. (By our convention on faces, we have $\\delta\\prec \\eta$.) The inclusion concerning the canonical compactifications of Bergman fans is denoted by $\\i^{\\eta\/\\delta}\\colon \\comp{\\Sigma^\\eta} \\hookrightarrow \\comp{\\Sigma^\\delta}$. The same simplifications hold.\n\n\\medskip\n\nWe also introduce projections $\\p_{\\delta}\\colon N^{\\sed(\\delta)}_\\mathbb{R} \\twoheadrightarrow \\TT\\delta$ as follows. Let $(\\,\\cdot\\,,\\cdot\\,)$ be an inner product on $N_\\mathbb{R}$. If $\\sed(\\delta)=\\conezero$, then we define $\\p_{\\delta}$ to be the orthogonal projection with respect to $(\\,\\cdot\\,,\\cdot\\,)$. We extend this definition to other strata as follows.\n\nFor any face $\\phi$ of sedentarity $\\sigma$, we observe that we can find a unique face $\\delta$ of sedentarity $\\conezero$ such that $\\phi=\\delta_\\infty^\\sigma$, and we define $\\p_\\phi$ as the quotient map\n\\[\\p_\\phi:=\\pi^{\\sigma}_*(\\p_\\delta),\\]\nmaking the following diagram commutative:\n\\[\n\\begin{tikzcd}\nN_\\mathbb{R} \\dar[\"\\pi^\\sigma\"] \\rar[\"\\p_\\delta\"] & \\TT\\delta \\dar[\"\\pi^\\sigma\"]  \\\\\nN^\\sigma_\\mathbb{R}=\\rquot{N_\\mathbb{R}}{N_{\\sigma, \\mathbb{R}}} \\rar[\"\\p_\\phi\"] &  \\TT\\phi = \\rquot{\\TT \\delta}{N_{\\sigma, \\mathbb{R}}}\n\\end{tikzcd} \\]\nIn addition, if $\\sed(\\eta)=\\sed(\\delta)$, then we define the relative projection map $\\p_{\\eta\/\\delta}$ by setting\n\\[\\p_{\\eta\/\\delta}:=\\p_\\delta\\rest{\\TT\\eta}.\n\\]\nIf $\\phi$ is a subface of $\\delta$ with the same sedentarity, we set $\\p_\\delta^\\phi := \\pi^\\phi_*(\\p_\\delta)$ and $\\p_{\\eta\/\\delta}^\\phi := \\pi^\\phi_*(\\p_{\\eta\/\\delta})$. We also set $\\cp_\\delta:=\\id-\\p_\\delta$ the orthogonal projection onto $\\ker(\\p_\\delta)$. We use the same notation for $\\cp_\\delta^\\phi$, etc.\n\nNotice that $\\pi^\\delta\\rest{\\ker(\\p_\\delta)}\\colon \\ker(\\p_\\delta) \\to N^\\delta_\\mathbb{R}$ is an isomorphism. To simplify the notation, we set $\\pi_\\delta^{-1}:=(\\pi^\\delta\\rest{\\ker(\\p_\\delta)})^{-1}$. In the same way, $\\pi_{\\eta\/\\delta}^{-1}:=(\\pi^{\\eta\/\\delta}\\rest{\\ker(\\p^\\delta_{\\eta})})^{-1}$.\n\n\\medskip\n\n\n\\subsubsection*{Operations on local cohomology groups}\n\nRecall that $\\i^{\\eta\/\\delta}$ induces a surjective map $\\i_{\\delta\\\/\\eta}^*$ from $H^\\bul(\\delta)$ to $H^\\bul(\\eta)$ of weight $0$, where $H^\\bul(\\delta):=H^\\bul(\\comp{\\Sigma^\\delta})$. The Poincar\u00e9 dual map of $\\i_{\\delta\\\/\\eta}^*$ is denoted $\\gys^{\\eta\/\\delta}$. It is an injective map of weight $2(\\dims\\eta-\\dims\\delta)$ from $H^\\bul(\\eta)$ to $H^\\bul(\\delta)$. We denote by $1^\\delta\\in H^0(\\delta)$ the corresponding unit. We denote by $x_{\\delta\/\\gamma} \\in A^1(\\delta)\\simeq H^2(\\delta)$ the element corresponding to the ray $\\delta^\\gamma$.\n\n\\medskip\n\n\n\\subsubsection*{Operations on faces}\nIf $\\phi$ and $\\psi$ are subfaces of a same face $\\delta$, we denote by $\\phi+\\psi$ the smallest face containing $\\phi$ and $\\psi$. When this has a meaning, we denote by $\\eta-\\delta$ the smallest face $\\phi$ such that $\\phi+\\delta=\\eta$. By $\\eta-\\delta+\\gamma$, for instance, we mean $(\\eta-\\delta)+\\gamma$. Abusing of the notation, by $\\p_{\\eta-\\delta+\\gamma\/\\gamma}$ we mean $\\p_{(\\eta-\\delta+\\gamma)\/\\gamma}$. We might sometimes write $\\p_{\\eta-\\delta\/\\gamma}$ instead of $\\p_{\\eta-\\delta+\\gamma\/\\gamma}$.\n\n\\medskip\n\n\n\\subsubsection*{Multivectors and multiforms $e$ and $\\nu$} We have a sign function $\\sign(\\,\\cdot\\,,\\cdot\\,)=\\textrm{sign}(\\,\\cdot\\,,\\cdot\\,)$ which verifies the diamond property\n\\[\\sign(\\gamma,\\delta)\\sign(\\delta,\\eta)=\\sign(\\gamma,\\delta')\\sign(\\delta',\\eta),\\]\nhere the faces are allowed to have different sedentarities, and $\\sign(v,e)=-\\sign(v',e)$ if $v$ and $v'$ are the two extremities of an edge $e$.\n\nLet $\\e_{\\delta\/\\gamma} \\in \\TT{\\delta^\\gamma}$ be the primitive element of the ray $\\delta^\\gamma$. Let $e_{\\delta\/\\gamma}:=\\sign(\\gamma,\\delta)\\e_{\\delta\/\\gamma}$. Let $s:=\\dims\\eta-\\dims\\delta$ and let $\\delta_0, \\delta_1, \\dots, \\delta_s$ be a flag of faces such that\n\\[ \\delta=\\delta_0\\ssubface\\delta_1\\ssubface\\cdots\\ssubface\\delta_s=\\eta. \\]\nWe define the multivector $e_{\\eta\/\\delta} \\in \\bigwedge^s\\TT{\\eta^\\delta}$ by\n\\[\ne_{\\eta\/\\delta} :=\\sign(\\delta_0,\\delta_1)\\dots\\sign(\\delta_{s-1},\\delta_s)\\e_{\\delta_1\/\\delta} \\wedge \\e_{(\\delta_2-\\delta_1)\/\\delta} \\wedge \\dots \\wedge \\e_{(\\delta_s-\\delta_{s-1})\/\\delta}.\n\\]\nThis definition is independent of the choice of the flag $\\delta_0,\\dots,\\delta_s $.\n\n\\medskip\n\nWe set $\\nvect_{\\delta\/\\gamma}:=\\pi_\\gamma^{-1}(\\e_{\\delta\/\\gamma})$ and $\\nu_{\\eta\/\\delta}:=\\pi_\\delta^{-1}(e_{\\eta\/\\delta})$. (Recall that by our convention, $\\pi_\\delta^{-1}$ denotes the inverse of the restriction of $\\pi_\\delta$ to the kernel of the projection map $\\p_\\delta$, which is an isomorphism.)\n\nIf $\\phi$ is a subface of $\\delta$ with the same sedentarity, we set $\\nu^\\phi_{\\eta\/\\delta}:=\\pi^\\phi(\\nu_{\\eta\/\\delta})$. In particular, $\\nu^\\delta_{\\eta\/\\delta}=e_{\\eta\/\\delta}$. Similarly, for a subface $\\phi$ of $\\gamma$ with the same sedentarity, we set $\\nvect^\\phi_{\\delta\/\\gamma}:=\\pi^\\phi(\\nvect_{\\delta\/\\gamma})$.\n\n\\smallskip\nLet $e_{\\eta\\\/\\delta}^\\dual\\in\\bigwedge^s\\TT^\\dual\\eta^\\delta$ be the dual of $e_{\\eta\/\\delta}$. Set $\\nu^\\dual_{\\delta\\\/\\eta}:=\\pi_\\delta^*(e^\\dual_{\\eta\\\/\\delta})\\in\\bigwedge^s\\TT^\\dual\\eta$ and $\\nu^{\\phi\\,*}_{\\eta\\\/\\delta}:=\\pi_{\\delta\/\\phi}^*(e^\\dual_{\\eta\\\/\\delta})\\in\\bigwedge^s\\TT^\\dual\\eta^\\phi$.\n\n\\smallskip\nWe also denote by $1_\\delta^\\dual\\in\\bigwedge^0\\TT^\\dual\\delta$ the corresponding unit.\n\n\\medskip\n\n\n\\subsubsection*{Operations on multivectors and on multiforms}\nLet $V$ is a vector space. Let $\\alpha\\in\\bigwedge^r V^\\dual$ be a multiform on $V$ and $\\u\\in\\bigwedge^s V$ be a multivector in $V$. Then we denote by $\\alpha \\vee \\u \\in \\bigwedge^{r-s}V^\\dual$ the \\emph{right contraction of $\\alpha$ by $\\u$}, i.e.,\n\\[ \\alpha\\vee\\u := \\begin{cases}\n  \\alpha(\\,\\cdot \\wedge \\u) & \\text{if $s\\geq r$}, \\\\\n  0 & \\text{otherwise}.\n\\end{cases} \\]\nWe also define the \\emph{left contraction of $\\alpha$ by $\\u$} by\n\\[ \\u\\vee\\alpha := \\begin{cases}\n  \\alpha(\\u \\wedge \\,\\cdot\\,) & \\text{if $s\\geq r$}, \\\\\n  0 & \\text{otherwise}.\n\\end{cases} \\]\n\nWe denote by $\\ker(\\alpha)$ the subspace of $V$ defined by\n\\[ \\ker(\\alpha) := \\{u \\in V \\st \\alpha\\vee u = 0 \\}. \\]\nIf $\\alpha$ is nonzero, the kernel is of dimension $\\dim(V)-r$. If $\\beta \\in \\bigwedge^t \\ker(\\alpha)^\\dual$ then we denote by $\\beta\\wedge\\alpha \\in \\bigwedge^{t+r}V^\\dual$ the multiform $\\~\\beta\\wedge\\alpha$ where $\\~\\beta$ is any extension of $\\beta$ to $\\bigwedge^t V$. The definition of $\\beta\\wedge\\alpha$ does not depend on the chosen extension.\n\n\n\n\\subsection{Useful facts}\nIn this section, we list some useful facts which will be used in the proofs of Propositions~\\ref{prop:tech_1} and~\\ref{prop:tech_2}.\n\n\\begin{prop} Using the conventions introduced in the previous section, the following holds.\n\\newcommand{\\ \\vspace*{-1em}}{\\ \\vspace*{-1em}}\n\\let\\olditem\\item\n\\renewcommand{\\item}{\\stepcounter{equation}\\olditem}\n\\begin{enumerate}[label=\\textnormal{(\\theequation)}, ref={\\theequation}]\n\\item \\label{com:eqn:maxsed_sed} If $\\sed(\\gamma)\\ssupface\\sed(\\delta)$ then\n  \\[ \\maxsed(\\gamma)=\\maxsed(\\delta). \\]\n\\item \\label{com:eqn:maxsed\/sed} \\ \\vspace*{-1em}\n  \\[ \\rquot{\\maxsed(\\delta)}{\\sed(\\delta)}=\\delta_\\infty. \\]\n\\item \\label{com:eqn:zeta_pi} If $\\sed(\\eta)=\\sed(\\delta)\\ssubface\\sed(\\gamma)$, then\n  \\[ \\zeta=\\eta_\\infty^{\\sed(\\gamma)}. \\]\n\\end{enumerate}\n\n\\bigskip\n\n\\noindent Let $\\ell$ and $\\ell'$ be two linear forms, $\\alpha$ be a multiform and $\\u$ and $\\v$ be two multivectors on a vector space $V$. the following holds.\n\\begin{enumerate}[resume*]\n\\item \\ \\vspace*{-1em}\n  \\[ \\alpha \\vee (\\u \\wedge \\v) = \\alpha \\vee \\v \\vee \\u. \\]\n\\item \\label{com:eqn:wedge_vee} If $u\\in\\ker(\\ell)$, then\n  \\[ \\alpha \\wedge \\ell \\vee u = -\\,\\alpha \\vee u \\wedge \\ell. \\]\n\\item \\label{com:eqn:wedge_dual} If $u'\\in\\ker(\\ell)$, then\n  \\[ (\\ell\\wedge\\ell')(u\\wedge u')=\\ell(u)\\cdot\\ell'(u'). \\]\n\n\\bigskip\n\n\\item \\label{com:eqn:gys_x} \\ \\vspace*{-1em}\n  \\[ \\i^*_{\\zeta\/\\eta}\\gys^{\\eta\/\\zeta}(x)=\\i^*_{\\zeta\/\\eta}\\gys^{\\eta\/\\zeta}(1^\\eta)x. \\]\n\\item \\label{com:eqn:i*_x} \\ \\vspace*{-1em}\n  \\[ \\gys^{\\eta\/\\zeta}\\i^*_{\\zeta\/\\eta}(x)=\\gys^{\\eta\/\\zeta}\\i^*_{\\zeta\/\\eta}(1^\\eta)x. \\]\n\\item \\label{com:eqn:gys} \\ \\vspace*{-1em}\n  \\[ \\gys^{\\eta\/\\zeta}(1^\\eta)=\\i^*_{\\gamma\\\/\\zeta}(x_{\\eta-\\zeta\/\\gamma}). \\]\n\n\\bigskip\n\n\\item \\label{com:eqn:nu} \\ \\vspace*{-1em}\n  \\[ \\nu_{\\eta\/\\phi} = \\nu_{\\delta\/\\phi} \\wedge \\nu_{\\eta\/\\delta}. \\]\n\\item \\label{com:eqn:nu*} \\ \\vspace*{-1em}\n  \\[ \\nu^\\dual_{\\phi\\\/\\eta} = \\nu^\\dual_{\\phi\\\/\\delta} \\wedge \\nu^\\dual_{\\delta\\\/\\eta}. \\]\n\\item \\label{com:eqn:pi_nu*} If $\\sed(\\gamma)=\\sed(\\delta)\\ssupface\\sed(\\delta')=\\sed(\\eta)$, then,\n  \\[ \\pi^{\\sed*}_{\\chi\/\\delta}(\\nu^\\dual_{\\gamma\\\/\\delta})=\\nu^\\dual_{\\delta'\\\/\\chi}. \\]\n\\item If $\\beta$ is a multiform on $N^\\delta_\\mathbb{R}$,\n  \\[ \\pi^{\\delta\\\/\\eta}_*(\\beta \\vee \\nu^\\delta_{\\eta\/\\delta}) \\text{ is well-defined.} \\]\n\\item If $\\phi$ is a subface of $\\delta$ of same sedentarity, then\n  \\[ \\cp_\\delta(\\nu_{\\eta\/\\phi}) = \\nu_{\\eta\/\\delta}. \\]\n\\item \\label{com:eqn:nu*_nu} \\ \\vspace*{-1em}\n  \\[ \\p_\\delta(\\nu_{\\delta'\/\\gamma})=\\nu^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nu_{\\delta'\/\\gamma}))\\nu_{\\delta\/\\gamma}. \\]\n\\item If $\\eta=\\delta+\\zeta$, even if $\\sed(\\delta)\\neq\\sed(\\gamma)$ (in which case $\\pi^{\\gamma\\\/\\delta}=\\id^{\\gamma\\\/\\delta}$),\n  \\[ \\pi^{\\gamma\\\/\\delta}(e_{\\eta\/\\delta}) = (-1)^{\\dims\\eta-\\dims\\delta}\\sign(\\gamma,\\delta)\\sign(\\zeta,\\eta)e_{\\zeta\/\\gamma}. \\]\n\n\\bigskip\n\n\\item \\label{com:eqn:p_nu*_nu} \\ \\vspace*{-1em}\n  \\[ \\p^*_{\\gamma\\\/\\delta}=(\\,\\cdot\\wedge\\nu^\\dual_{\\gamma\\\/\\delta})\\vee\\nu_{\\delta\/\\gamma}. \\]\n\\item \\ \\vspace*{-1em}\n  \\[ \\p_\\delta \\p_\\eta=\\p_\\delta \\p_\\eta = \\p_\\delta \\quad\\text{ and }\\quad \\cp_\\delta \\p_\\eta=\\p_\\eta \\cp_\\delta. \\]\n\\item \\label{com:eqn:p_pi} If $\\sed(\\gamma)=\\sed(\\delta)\\ssupface\\sed(\\delta')=\\sed(\\eta)$, then\n  \\[ \\pi^{\\sed}_{\\delta'\/\\gamma}\\p_{\\chi\/\\delta'} = \\p_{\\delta\/\\gamma}\\pi^{\\sed}_{\\chi\/\\delta}. \\]\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof} We omit the proof which can be obtained by straightforward computations.\n\\end{proof}\n\n\n\n\\subsection{Proof of Proposition~\\ref{prop:tech_1}}\n\\label{sec:com:trop}\n\nIn this section, we define a (non canonical) isomorphism $\\Phi\\colon \\Tot^\\bul(\\AA_\\trop^{\\bul,\\bul}) \\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} C_\\trop^{p,\\bul}$. We then prove that we can choose two differentials $\\dfrak_\\trop$ and $\\d_\\trop$ of bidegree $(1,0)$ and $(0,1)$ on $\\AA_\\trop^{\\bul, \\bul}$ such that we have the equality $\\dfrak_\\trop+\\d_\\trop = \\Phi^{-1} \\partial_\\trop \\Phi$ where $\\partial_\\trop$ is the usual tropical differential on $C_\\trop^{p,\\bul}$. The two differentials we obtain are those we have already seen in Section \\ref{sec:steenbrink}. This will prove Proposition~\\ref{prop:tech_1}.\n\n\\medskip\n\nRecall that, if $\\dims\\delta = a+b$,\n\\[ \\AA_\\trop^{a,b}[\\delta] = \\bigwedge^b\\TT^\\dual\\delta \\otimes \\SF^{p-b}(\\conezero^\\delta)  \\quad\\text{ and }\\quad C_\\trop^{p,\\dims\\delta}[\\delta] = \\SF^p(\\delta), \\]\nand other pieces are zero. The differential $\\partial_\\trop$ is equal to the sum $\\iota^*_\\trop+\\pi^*_\\trop$ where\n\\[ \\iota^*_\\trop \\bigl[C_\\trop^{p,\\dims\\gamma}[\\gamma] \\to C_\\trop^{p,\\dims\\delta}[\\delta]\\bigr] = \\sign(\\gamma,\\delta)\\iota^*_{\\gamma\\\/\\delta} \\]\nand, if $\\sed(\\gamma)\\ssupface\\sed(\\delta)$, then\n\\[ \\pi^*_\\trop \\bigl[C_\\trop^{p,\\dims\\gamma}[\\gamma] \\to C_\\trop^{p,\\dims\\delta}[\\delta]\\bigr] = \\sign(\\gamma,\\delta)\\pi^*_{\\gamma\\\/\\delta}. \\]\n\n\\medskip\n\nRecall as well from Section~\\ref{sec:steenbrink} that we have a filtration $W^\\bul$ on $C_\\trop^{p,\\bul}$. With our notations introduced above, this is defined by\n\\[ W^sC_\\trop^{p,\\dims\\delta}[\\delta] := \\Bigl\\{\\, \\alpha\\in\\SF^p(\\delta) \\st \\forall \\u\\in\\bigwedge^{\\dims\\delta-s+1}\\TT_\\delta, \\alpha\\vee\\u=0 \\Bigr\\}. \\]\nWe proved in Section~\\ref{sec:steenbrink} that this filtration is preserved by $\\partial_\\trop$.\n\n\\medskip\n\nBy an abuse of the notation, we also denote by $W^\\bul$ the filtration on $\\AA_\\trop^{\\bul,\\bul}$ induced by the first index.\n\n\\medskip\n\nWe now define $\\Phi\\colon\\AA_\\trop^{\\bul,\\bul} \\to C_\\trop^{p,\\bul}$ by\n\\[ \\Phi\\bigl[\\AA_\\trop^{a,b}[\\delta] \\to C_\\trop^{p,a+b}[\\delta]\\bigr] = \\p^*_\\delta \\wedge \\pi^*_\\delta\\]\nif $a+b=\\dims\\delta$, and all other pieces are zero. Set\n\\[ \\Phi_a := \\Phi[\\AA_\\trop^{a,\\bul} \\to *]. \\]\n\nWe define $\\Psi\\colon C_\\trop^{p,\\bul} \\to \\AA_\\trop^{\\bul,\\bul}$ by\n\\[ \\Psi\\bigl[C_\\trop^{p,a+b}[\\delta] \\to \\AA_\\trop^{a,b}[\\delta]\\bigr](\\alpha)(\\u \\otimes \\v) = \\alpha(\\u \\wedge \\pi^{-1}_\\delta(\\v)) \\]\nif $a+b=\\dims\\delta$, and other pieces are zero. Set\n\\[ \\Psi_a := \\Psi[* \\to \\AA_\\trop^{a,\\bul}]. \\]\n\n\\begin{prop}\nThe map $\\Phi$ and $\\Psi$ are inverse isomorphisms of weight zero, and both preserve the filtration $W^\\bul$. Moreover, for any $a$ and $k$, $\\Psi_{a+k}\\partial_\\trop\\Phi_a = 0$ is zero except for $k\\in\\{0,1\\}$.\n\\end{prop}\n\nSet $\\d_\\trop := \\bigoplus_{a}\\Psi_a \\partial_\\trop \\Phi_a$ and $\\dfrak_\\trop := \\d^\\i_\\trop + \\d^\\pi_\\trop$ where\n\\[ \\d^\\i_\\trop := \\Psi_{a+1} \\iota^*_\\trop \\Phi_a \\quad\\text{ and }\\quad \\d^\\pi_\\trop := \\Psi_{a+1} \\pi^*_\\trop \\Phi_a. \\]\nBy the previous proposition it is now clear that $\\Phi\\colon (\\Tot^\\bul(\\AA_\\trop^{\\bul,\\bul}), \\dfrak_\\trop + \\d_\\trop) \\xrightarrow{\\raisebox{-3pt}[0pt][0pt]{\\small$\\hspace{-1pt}\\sim$}} (C_\\trop^{p,\\bul}, \\partial_\\trop)$ is an isomorphism of complexes which respects the filtrations. It remains to give an explicit formula of $\\d_\\trop$ and $\\dfrak_\\trop$. This can be shown by direct computations.\n\n\\begin{prop}\nIf $\\dims\\delta=a+b$, then\n\\[ \\d_\\trop\\bigl[\\AA_\\trop^{a,b-1}[\\gamma] \\to \\AA_\\trop^{a,b}[\\delta]\\bigr](\\alpha \\otimes \\beta) = \\sign(\\gamma,\\delta)(\\alpha \\wedge \\nu^\\dual_{\\gamma\\\/\\delta}) \\otimes (\\pi^{\\gamma\\\/\\delta}_*(e_{\\delta\/\\gamma} \\vee \\beta)), \\]\nas stated in Proposition \\ref{prop:grading}, and\n\\[ \\d^\\i_\\trop\\bigl[\\AA_\\trop^{a-1,b}[\\gamma] \\to \\AA_\\trop^{a,b}[\\delta]\\bigr] = \\begin{cases}\n  \\sign(\\gamma, \\delta)\\p_{\\gamma\\\/\\delta}^* \\otimes (\\pi_{\\delta\/\\gamma}^{-1})^*, & \\text{if $\\sed(\\gamma)=\\sed(\\delta)$,} \\\\\n  0 & \\text{otherwise,}\n\\end{cases} \\]\nas stated by Equation \\eqref{eqn:d''}. Moreover,\n\\[ \\d^\\pi_\\trop\\bigl[\\AA_\\trop^{a-1,b}[\\gamma] \\to \\AA_\\trop^{a,b}[\\delta]\\bigr] = \\begin{cases}\n  \\sign(\\gamma, \\delta)\\pi^{\\sed*}_{\\gamma\\\/\\delta} \\otimes \\id & \\text{if $\\sed(\\gamma) \\ssupface \\sed(\\delta)$,} \\\\\n  0 & \\text{otherwise.}\n\\end{cases} \\]\n\\end{prop}\n\nIt thus follows that $\\d_\\trop$ and $\\dfrak_\\trop$ defined above coincide with $\\d_\\trop$ and $\\dfrak_\\trop$ of Section~\\ref{sec:steenbrink}.\n\n\n\n\\subsection{Differentials of $\\AA^{\\bul,\\bul,\\bul}$}\n\\label{sec:maps}\n\nWe now recall the definition of the differentials on $\\AA^{\\bul,\\bul,\\bul}$. Recall that, for $\\dims\\delta=a+b$ and $\\dims\\eta=p+a-b'$,\n\\begin{align*}\n\\AA_\\trop^{a,b}[\\delta] &= \\bigwedge^b\\TT^\\dual\\delta \\otimes \\SF^{p-b}(\\conezero^\\delta), \\\\\n\\AA_\\ST^{a,b'}[\\eta] &= H^{2b'}(\\eta)\\text{ if $\\dims{\\eta_\\infty} \\geq a$ and $b'\\geq p$, and} \\\\\n\\AA^{a,b,b'}_\\Dnop[\\delta,\\eta] &= \\bigwedge^b\\TT^\\dual\\delta \\otimes H^{2b'}(\\eta).\n\\end{align*}\n\nWhen the domain or the codomain of a map is clear, we only indicate the concerned faces. For $\\d^\\pi$, we assume that $\\sed(\\gamma)=\\sed(\\zeta)\\ssupface\\sed(\\delta)=\\sed(\\eta)$. We have\n\\begin{align*}\n\\d_\\trop[\\gamma \\to \\delta]\n  &= \\sign(\\gamma,\\delta)(\\,\\cdot \\wedge \\nu^\\dual_{\\gamma\\\/\\delta}) \\otimes \\pi^{\\gamma\\\/\\delta}_*(e_{\\delta\/\\gamma} \\vee \\cdot\\,), \\\\\n\\d'_\\trop\\bigl[\\AA_\\trop^{a,b}[\\delta] \\to \\delta,\\eta \\bigr]\n  &= (-1)^{(a+1)(b+p)+\\dims{\\sed(\\delta)}}\\id \\otimes (\\,\\cdot \\vee e_{\\eta\/\\delta})1_\\eta, \\\\\n\\d^\\i_\\trop[\\gamma \\to \\delta]\n  &= \\sign(\\gamma, \\delta)\\p_{\\gamma\\\/\\delta}^* \\otimes (\\pi_{\\gamma\\\/\\delta}^{-1})^*, \\\\\n\\d^\\pi_\\trop[\\gamma \\to \\delta]\n  &= \\sign(\\gamma, \\delta)\\pi^{\\sed*}_{\\gamma\\\/\\delta} \\otimes \\id^{\\gamma\\\/\\delta},\n\\end{align*}\n\\begin{align*}\n\\d_\\ST\\bigl[\\AA_\\ST^{a,b'}[\\eta] \\to \\delta,\\eta\\bigr]\n  &= (-1)^{a+b'}1^\\dual_\\delta \\otimes \\id  \\quad\\text{if $\\maxsed(\\delta)=\\maxsed(\\eta)$}, \\\\\n\\d'_\\ST[\\mu \\to \\eta]\n  &= \\sign(\\eta,\\mu)\\gys^{\\mu\/\\eta}  \\quad\\text{if $\\maxsed(\\eta)=\\maxsed(\\mu)$}, \\\\\n\\d^\\i_\\ST[\\eta \\to \\mu]\n  &= \\sign(\\eta,\\mu) \\i^*_{\\eta\\\/\\mu}  \\quad\\text{if $\\maxsed(\\eta)=\\maxsed(\\mu)$}, \\\\\n\\d^\\pi_\\ST[\\eta \\to \\mu]\n  &= \\sign(\\eta,\\mu)\\id^{\\eta\\\/\\mu}  \\quad\\text{(by \\eqref{com:eqn:maxsed_sed}, $\\maxsed(\\eta)=\\maxsed(\\mu)$)},\n\\end{align*}\n\\begin{align*}\n\\d_\\Dnop\\bigl[\\AA^{a,b,b'}[\\gamma,\\eta] \\to \\delta,\\eta\\bigr]\n  &= (-1)^{a+b'}\\sign(\\gamma,\\delta)(\\,\\cdot \\wedge \\nu^\\dual_{\\gamma\\\/\\delta}) \\otimes \\id, \\\\\n\\d'_\\Dnop[\\delta,\\mu \\to \\delta,\\eta]\n  &= \\sign(\\eta,\\mu)\\id \\otimes \\gys^{\\mu\/\\eta}, \\\\\n\\d^\\i_\\Dnop[\\gamma,\\zeta \\to \\delta,\\eta]\n  &= \\sign(\\zeta,\\eta) \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta\/\\gamma}))\\p_{\\gamma\\\/\\delta}^* \\otimes \\i^*_{\\zeta\\\/\\eta}, \\\\\n\\d^\\pi_\\Dnop[\\gamma,\\zeta \\to \\delta,\\eta]\n  &= \\sign(\\zeta,\\eta)\\pi^{\\sed*}_{\\gamma\\\/\\delta} \\otimes \\id^{\\zeta\\\/\\eta}.\n\\end{align*}\n\n\n\n\\subsection{Proof that $\\partial=\\dfrak+\\d+\\d'$ is a differential} Let $\\partial = \\dfrak+\\d+\\d'$ with $\\dfrak=\\d^\\i+\\d^\\pi$. In this section, we prove that $\\partial\\circ\\partial=0$. To do so, we separate thirty cases summarized in the following table. For example, the first case (denoted $a$ in the top-left corner) concerns the proof that $(\\d^2)[\\AA_\\trop^{\\bul,\\bul} \\to *]=0$. The case just after is the proof that $(\\d\\d'+\\d'\\d)[\\AA_\\trop^{\\bul,\\bul} \\to *]=0$, etc. Some cases has been gathered (in particular the easy cases and those that have been already done). In each case, we fix a context by precising the faces involved. We let the reader check that the studied case is general: no other configuration of faces can be involved.\n\n{\n\\newcommand\\taa{\\ref{com:trop}}\n\\newcommand\\tab{\\ref{com:tab}}\n\\newcommand\\tac{\\ref{com:trop}}\n\\newcommand\\tad{\\ref{com:trop}}\n\\newcommand\\tbb{\\ref{com:tbb}}\n\\newcommand\\tbc{\\ref{com:tbc}}\n\\newcommand\\tbd{\\ref{com:tbd}}\n\\newcommand\\tcc{\\ref{com:trop}}\n\\newcommand\\tcd{\\ref{com:trop}}\n\\newcommand\\tdd{\\ref{com:trop}}\n\n\\newcommand\\saa{\\ref{com:saa}}\n\\newcommand\\sab{\\ref{com:ST_pi}}\n\\newcommand\\sac{\\ref{com:sac}}\n\\newcommand\\sad{\\ref{com:ST_pi}}\n\\newcommand\\sbb{\\ref{com:ST}}\n\\newcommand\\sbc{\\ref{com:ST}}\n\\newcommand\\sbd{\\ref{com:ST_pi}}\n\\newcommand\\scc{\\ref{com:ST}}\n\\newcommand\\scd{\\ref{com:ST_pi}}\n\\newcommand\\sdd{\\ref{com:ST_pi}}\n\n\\newcommand\\aaa{\\ref{com:a_easy}}\n\\newcommand\\aab{\\ref{com:a_easy}}\n\\newcommand\\aac{\\ref{com:aac}}\n\\newcommand\\aad{\\ref{com:a_easy}}\n\\newcommand\\abb{\\ref{com:abb}}\n\\newcommand\\abc{\\ref{com:abc},\\ref{com:abc'},\\ref{com:abc''}}\n\\newcommand\\abd{\\ref{com:a_easy}}\n\\newcommand\\acc{\\ref{com:acc}}\n\\newcommand\\acd{\\ref{com:acd}}\n\\renewcommand\\add{\\ref{com:a_easy}}\n\\[ \\bigarray \\]\n}\n\n\\begin{enumerate}[label={\\bf(\\alph*)}, ref=\\alph*, leftmargin=0pt]\n\\newcommand{\\Longleftarrow\\quad}{\\Longleftarrow\\quad}\n\\newcommand{\\Longrightarrow\\quad}{\\Longrightarrow\\quad}\n\\newcommand{\\Longleftrightarrow\\quad}{\\Longleftrightarrow\\quad}\n\n\\item \\label{com:trop}$(\\d_\\trop+\\dfrak_\\trop)^2=0$. This was already done in Section \\ref{sec:com:trop}.\n\n\\medskip\n\n\\item \\label{com:tab} $\\d\\d'+\\d'\\d =0$ in $\\AA^{\\bul, \\bul}_\\trop$. We need to show that for integers $a,b$, a face $\\gamma$ with $\\dims \\gamma =a+b$, and $\\alpha \\otimes \\beta \\in\\bigwedge^b\\TT^\\dual\\gamma \\otimes \\SF^{p-b}(\\conezero^\\gamma)$, we have $(\\d\\d'+\\d'\\d)\\bigl[\\AA_\\trop^{a,b}[\\gamma] \\to \\AA^{a,b,0}[\\delta,\\eta]\\bigr](\\alpha\\otimes\\beta) = 0$. This is obtained by the following chain of implications.\n\\begin{align*}\n& \\hspace{-2cm}\\nu^\\gamma_{\\eta\/\\gamma}\n  = \\nu^\\gamma_{\\delta\/\\gamma} \\wedge \\nu^\\gamma_{\\eta\/\\delta}\\\\\n\\Longrightarrow\\quad& (-1)^a\\beta\\vee e_{\\eta\/\\gamma}\n  = -(-1)^{a+1}(e_{\\delta\/\\gamma} \\vee \\beta) \\vee \\pi_{\\delta\/\\gamma}^{-1}(e_{\\eta\/\\delta})\\\\\n\\Longrightarrow\\quad& (-1)^{a+0}\\sign(\\gamma,\\delta)(-1)^{(a+1)(b+p)+\\dims{\\sed(\\gamma)}}\\alpha\\vee\\nu^\\dual_{\\gamma\\\/\\delta}\\otimes (\\beta\\vee e_{\\eta\/\\gamma})1_\\eta\n  \\\\&\\qquad+ (-1)^{(a+1)(b+1+p)+\\dims{\\sed(\\delta)}}\\sign(\\gamma,\\delta)\\alpha\\vee\\nu^\\dual_{\\gamma\\\/\\delta} \\otimes (\\pi^{\\gamma\\\/\\delta}_*(e_{\\delta\/\\gamma} \\vee \\beta) \\vee e_{\\eta\/\\delta}) 1_\\eta = 0,\n\\end{align*}\nwhich is the desired equality given the differentials.\n\n\\medskip\n\n\\item \\label{com:tbb} We have $\\d'\\circ \\d'=0$ in $\\AA^{\\bul, \\bul}_\\trop$. Let $a, b$ be two integers, let $\\delta$ be a face of dimension $\\dims\\delta =a+b$, and choose $\\alpha\\otimes \\beta \\in\\bigwedge^b\\TT^\\dual\\delta \\otimes \\SF^{p-b}(\\conezero^\\delta)$.\nThen we have to prove that $(\\d'^2)\\bigl[\\AA_\\trop^{a,b}[\\delta] \\to \\AA^{a,b,1}[\\delta,\\eta]\\bigr](\\alpha\\otimes\\beta) = 0$. We have, using the linear relations in the cohomology ring $H^\\bul(\\eta)$ via the isomorphism with the Chow ring of $\\Sigma^\\eta$,\n\\begin{align*}\n& \\hspace{-2cm}\\ssum_{\\mu'} \\bigl(\\pi^{\\delta\\\/\\eta}_*(\\beta \\vee \\nu^\\delta_{\\eta\/\\delta})\\bigr)(\\e_{\\mu'\/\\eta}) x_{\\mu'\/\\eta} = 0 \\\\\n\\Longrightarrow\\quad& (-1)^{\\dims\\eta-\\dims\\delta}\\ssum_{\\mu'} (\\beta \\vee \\nu^\\delta_{\\eta\/\\delta} \\vee \\nvect^\\delta_{\\mu'\/\\eta}) x_{\\mu'\/\\eta} = 0 \\\\\n\\Longrightarrow\\quad& \\ssum_{\\mu'} \\sign(\\eta,\\mu') (\\beta \\vee \\nu^\\delta_{\\mu'\/\\eta} \\vee \\nu^\\delta_{\\eta\/\\delta}) x_{\\mu'\/\\eta} = 0 \\\\\n\\Longrightarrow\\quad& \\sum_{\\mu'\\ssupface\\eta \\\\ \\sed(\\mu')=\\sed(\\eta)} \\sign(\\eta,\\mu')(-1)^{(b+p)(a+1)+\\dims{\\sed(\\delta)}} \\alpha \\otimes (\\beta \\vee e_{\\mu'\/\\delta}) \\gys^{\\mu'\/\\eta}(1_\\mu') = 0.\n\\end{align*}\n\n\\medskip\n\n\\item \\label{com:tbc} We have to show that $\\d'\\d^\\i+\\d^\\i\\d'=0$ in $\\AA^{\\bul, \\bul}_\\trop$. Let $a, b$ be two integers, let $\\gamma$ be a face of dimension $\\dims\\gamma =a+b$, and choose $\\alpha\\otimes \\beta \\in\\bigwedge^b\\TT^\\dual\\gamma \\otimes \\SF^{p-b}(\\conezero^\\gamma)$. We need to prove\n\\[(\\d'\\d^\\i+\\d^\\i\\d')[\\AA_\\trop^{a,b}[\\gamma] \\to \\AA^{a+1,b,0}[\\delta,\\eta]](\\alpha \\otimes \\beta) = 0.\\]\nThis is equivalent to\n\\begin{align*}\n&\\Longleftrightarrow\\quad \\sign(\\gamma,\\delta) (-1)^{(a+2)(b+p)+\\dims{\\sed(\\delta)}} \\p^*_{\\gamma\\\/\\delta}(\\alpha) \\otimes ((\\pi_{\\gamma\\\/\\delta}^{-1})^*(\\beta) \\vee e_{\\eta\/\\delta})1_\\eta\n  \\\\&\\quad + \\sum_{\\zeta' \\ssubface \\eta \\\\ \\zeta' \\succ \\gamma} \\sign(\\zeta',\\eta) (-1)^{(a+1)(b+p)+\\dims{\\sed(\\gamma)}} \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta'\/\\gamma})) \\p^*_{\\gamma\\\/\\delta}(\\alpha) \\otimes (\\beta \\vee e_{\\zeta'\/\\gamma}) \\i^*_{\\zeta'\\\/\\eta}(1_{\\zeta'}) =0 \\\\\n&\\Longleftrightarrow\\quad \\sign(\\gamma,\\delta) (-1)^{p-(b+1)} \\pi_{\\delta\/\\gamma}^{-1}(e_{\\eta\/\\delta})\n  = {\\ssum}_{\\zeta'} \\sign(\\zeta',\\eta) \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta'\/\\gamma})) e_{\\zeta'\/\\gamma} \\\\\n&\\Longleftrightarrow\\quad \\nu_{\\eta\/\\delta}\n  = \\sign(\\gamma,\\delta) (-1)^{\\dims\\eta-\\dims\\delta} {\\ssum}_{\\zeta'} \\sign(\\zeta',\\eta) \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta'\/\\gamma})) \\nu_{\\zeta'\/\\gamma}.\n\\end{align*}\n\nThus, we reduce to proving the last equation. Let $s=\\dims\\eta-\\dims\\delta$. Let $\\delta_0,\\dots,\\delta_s$ such that\n\\[ \\delta=\\delta_0\\ssubface\\delta_1\\ssubface\\cdots\\ssubface\\delta_s=\\eta. \\]\nSet $\\epsilon=\\delta-\\gamma$, $\\gamma_i=\\delta_i-\\epsilon$ and $\\zeta_i=\\eta-(\\delta_{i+1}-\\delta_{i})=\\eta-(\\gamma_{i+1}-\\gamma_i)$. Then\n\\begin{align*}\n\\nu_{\\eta\/\\delta}\n  &= \\oldbigwedge_{i=0}^{s-1} \\sign(\\delta_i,\\delta_{i+1}) \\nvect_{\\delta_{i+1}-\\delta_i\/\\delta} \\\\\n  &= \\bigwedge_{\\,i} -\\sign(\\gamma_i,\\delta_i)\\sign(\\gamma_i,\\gamma_{i+1})\\sign(\\gamma_{i+1},\\delta_{i+1}) \\cp_\\delta(\\nvect_{\\gamma_{i+1}-\\gamma_i\/\\gamma}) \\\\\n  &= (-1)^s\\sign(\\gamma_0,\\delta_0)\\sign(\\gamma_s,\\delta_s) \\bigwedge_{\\,i} \\sign(\\gamma_i,\\gamma_{i+1}) \\bigl(\\nvect_{\\gamma_{i+1}-\\gamma_i\/\\gamma} - \\p_\\delta(\\nvect_{\\gamma_{i+1}-\\gamma_i\/\\gamma})\\bigr).\n\\end{align*}\nBy \\eqref{com:eqn:nu*_nu},\n\\[ \\p_\\delta(\\nvect_{\\gamma_{i+1}-\\gamma_i\/\\gamma})=\\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta_i}))\\nvect_{\\delta\/\\gamma}. \\]\nHence,\n\\begin{align*}\n&\\nu_{\\eta\/\\delta} = (-1)^s\\sign(\\gamma,\\delta)\\sign(\\eta-\\epsilon,\\eta) \\Bigl( \\bigwedge_{\\,i} \\sign(\\gamma_i,\\gamma_{i+1}) \\nvect_{\\gamma_{i+1}-\\gamma_i\/\\gamma} \\\\\n&\\qquad- \\sum_i (-1)^{s-i-1}\\sign(\\gamma_i,\\gamma_{i+1})\\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta_i\/\\gamma}))\\oldbigwedge_{\\substack{j=0 \\\\ j\\neq i}}^{s-1} \\sign(\\gamma_j,\\gamma_{j+1}) \\nvect_{\\gamma_{j+1}-\\gamma_j\/\\gamma}\\ \\wedge \\nvect_{\\delta\/\\gamma}  \\Bigr).\n\\end{align*}\nIn the big brackets, the first wedge product is simply $\\nu_{\\gamma_s\/\\gamma_0}=\\nu_{\\eta-\\epsilon\/\\gamma}$. For the sum, set $\\kappa_i:=\\gamma_{i+1}-\\gamma_i$. We get\n\\begin{align*}\n&\\sign(\\gamma_j,\\gamma_{j+1}) \\nvect_{\\gamma_{j+1}-\\gamma_j\/\\gamma} =  \\\\\n&\\qquad-\\sign(\\gamma_j-\\kappa_i,\\gamma_j)\\sign(\\gamma_j-\\kappa_i,\\gamma_{j+1}-\\kappa_i)\\sign(\\gamma_{j+1}-\\kappa_i,\\gamma_{j+1}) \\nvect_{(\\gamma_{j+1}-\\kappa_i)-(\\gamma_j-\\kappa_i)\/\\gamma}.\n\\end{align*}\nHence,\n\\begin{align*}\n&\\oldbigwedge_{\\substack{j=i+1}}^{s-1} \\sign(\\gamma_j,\\gamma_{j+1}) \\nvect_{\\gamma_{j+1}-\\gamma_j\/\\gamma} \\\\\n  &\\qquad= (-1)^{s-i-1}\\sign(\\gamma_{i+1}-\\kappa_i,\\gamma_{i+1})\\sign(\\gamma_s-\\kappa_i,\\gamma_s)\\nu_{\\gamma_s-\\kappa_i\/\\gamma_{i+1}-\\kappa_i} \\\\\n  &\\qquad= (-1)^{s-i-1}\\sign(\\gamma_i,\\gamma_{i+1})\\sign(\\zeta_i-\\epsilon,\\eta-\\epsilon)\\nu_{\\zeta_i-\\epsilon\/\\gamma_i}.\n\\end{align*}\nThus, the $i$-th term of the sum equals\n\\begin{align*}\n& \\sign(\\zeta_i-\\epsilon,\\eta-\\epsilon) \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta_i})) \\nu_{\\gamma_i\/\\gamma} \\wedge \\nu_{\\zeta_i-\\epsilon\/\\gamma_i} \\wedge \\nvect_{\\delta\/\\gamma} \\\\\n  &\\qquad = \\sign(\\zeta_i-\\epsilon,\\eta-\\epsilon) \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta_i\/\\gamma})) \\nu_{\\zeta_i-\\epsilon\/\\gamma} \\wedge \\sign(\\zeta_i-\\epsilon,\\zeta_i) \\nu_{\\zeta_i\/\\zeta_i-\\epsilon} \\\\\n  &\\qquad = -\\sign(\\zeta_i,\\eta)\\sign(\\eta-\\epsilon,\\eta) \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta_i\/\\gamma}))\\nu_{\\zeta_i\/\\gamma}.\n\\end{align*}\nTherefore,\n\\begin{align*}\n\\nu_{\\eta\/\\delta}\n  &= (-1)^s\\sign(\\gamma,\\delta) \\Bigl(\\sign(\\eta-\\epsilon,\\eta)\\nu_{\\eta-\\epsilon\/\\gamma} + \\sum_i \\sign(\\zeta_i,\\eta)\\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta_i}))\\nu_{\\zeta_i\/\\gamma} \\Bigr).\n\\end{align*}\nNotice that the $\\zeta_i$ are exactly the faces $\\zeta'$ such that $\\delta\\prec\\zeta'\\ssubface\\eta$, and $\\eta-\\epsilon$ is the only face $\\zeta'$ such that $\\gamma\\prec\\zeta'\\ssubface\\eta$ and such that $\\delta\\not\\prec\\zeta'$. Moreover,\n\\[ \\nu^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-(\\eta-\\epsilon)\/\\gamma)})) = \\nu^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\delta\/\\gamma)})) = 1. \\]\nFinally,\n\\[ \\nu_{\\eta\/\\delta}\n  = (-1)^s\\sign(\\gamma,\\delta) \\sum_{\\zeta' \\ssubface \\eta \\\\ \\zeta' \\succ \\gamma} \\sign(\\zeta',\\eta) \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta'\/\\gamma})) \\nu_{\\zeta'\/\\gamma},\\]\nwhich is what we wanted to prove.\n\n\\medskip\n\n\\item \\label{com:tbd} We have to show that $\\d'\\d^\\pi+\\d^\\pi\\d'=0$ in $\\AA^{\\bul,\\bul}_\\trop$. Let $a, b$ be two integers, let $\\gamma$ be a face of dimension $\\dims\\gamma =a+b$, and choose $\\alpha\\otimes \\beta \\in\\bigwedge^b\\TT^\\dual\\gamma \\otimes \\SF^{p-b}(\\conezero^\\gamma)$. Suppose moreover that $\\sed(\\eta)=\\sed(\\delta)\\ssubface\\sed(\\gamma)$. Let $\\zeta=\\eta_\\infty^{\\sed(\\gamma)}$. Then, we have\n\\begin{align*}\n&\\hspace{-2cm}(\\d'\\d^\\pi+\\d^\\pi\\d')\\bigl[\\AA_\\trop^{a,b}[\\gamma] \\to \\AA^{a,b,0}[\\delta, \\eta]\\bigr](\\alpha \\otimes \\beta)=0\\\\\n\\Longleftrightarrow\\quad& (-1)^{(a+2)(b+p)+\\dims{\\sed(\\delta)}}\\sign(\\gamma,\\delta)\\pi^{\\sed*}_{\\gamma\\\/\\delta}(\\alpha)\\otimes(\\id^{\\gamma\\\/\\delta}(\\beta) \\vee e_{\\eta\/\\delta})1_\\eta\n  \\\\&\\qquad+ \\sign(\\zeta,\\eta)(-1)^{(a+1)(b+p)+\\dims{\\sed(\\gamma)}}\\pi^{\\sed *}_{\\gamma\/\\delta}(\\alpha) \\otimes (\\beta \\vee e_{\\zeta\/\\gamma})\\id^{\\zeta\\\/\\eta}(1_{\\zeta}) = 0, \\\\\n\\Longleftrightarrow\\quad& (-1)^{b+p+1}\\sign(\\gamma,\\delta)\\id^{\\delta\/\\gamma}(e_{\\eta\/\\delta})\n  = -\\sign(\\zeta,\\eta)e_{\\zeta\/\\gamma}, \\\\\n\\Longleftrightarrow\\quad& \\id^{\\delta\/\\gamma}(e_{\\eta\/\\delta})\n  = (-1)^{\\dims\\eta-\\dims\\delta}\\sign(\\gamma,\\delta)\\sign(\\zeta,\\eta)e_{\\zeta\/\\gamma},\n\\end{align*}\nwhich follows from the definition.\n\n\\medskip\n\n\\item \\label{com:ST_pi} We need to show that $(\\d^\\pi_\\ST)^2=\\d^\\i\\d^\\pi_\\ST+\\d^\\pi\\d^\\i_\\ST=\\d'\\d^\\pi_\\ST+\\d^\\pi\\d'_\\ST=\\d\\d^\\pi_\\ST+\\d^\\pi\\d_\\ST=\\d^\\i\\d'_\\ST+\\d'\\d^\\i_\\ST=0$ in $\\AA^{\\bul, \\bul}_\\ST$. This is clear by looking at the definitions and using \\eqref{com:eqn:zeta_pi} and the defining property of the sign function.\n\n\\medskip\n\n\\item \\label{com:ST} We have to show that $(\\d'+\\d^\\i)^2\\bigl[\\AA^{a,b'}_\\ST[\\eta] \\to *]=0$.\n\nAssume $\\sigma=\\sed\\eta$ and $\\xi=\\maxsed\\eta$. If $\\sigma=\\xi$ the statement holds by applying Proposition \\ref{prop:app1} to $(X^\\sigma)_\\f$. Otherwise, we can reduce to the previous case as follows. By \\eqref{com:ST_pi}, the two differentials $\\d'_\\ST$ and $\\d^\\i_\\ST$ anticommute with $\\d^\\pi_\\ST$. Recall that $\\d^\\pi_\\ST$ is, up to a sign, identity on each piece. Thus, setting\n\\[ \\partial:=(\\d^\\pi)^{\\dims\\xi-\\dims\\sigma}[\\AAa{\\substack{\\sed=\\xi \\\\ \\maxsed=\\xi}}_\\ST^{\\bul,\\bul}\\to\\AAa{\\substack{\\sed=\\sigma \\\\ \\maxsed=\\xi}}_\\ST^{\\bul,\\bul}], \\]\nwe get\n\\[ \\partial^{-1}(\\d'+\\d^\\i)^2\\partial=(-1)^{\\dims\\xi-\\dims\\sigma}(\\d'+\\d^\\i)^2[\\AAa{\\substack{\\sed=\\xi \\\\ \\maxsed=\\xi}}_\\ST^{\\bul,\\bul}\\to *]. \\]\n\n\\medskip\n\n\\item \\label{com:saa} This says that $(\\d^2)[\\AA_\\ST^{a,b'}[\\eta] \\to \\AA^{a,1,b'}[\\delta,\\eta]](x) = 0.$ This is equivalent to\n\\begin{align*}\n\\Longleftrightarrow\\quad& \\sum_{\\gamma'\\ssubface\\delta \\\\ \\maxsed(\\gamma')=\\maxsed(\\eta) \\\\ \\sed(\\gamma')=\\sed(\\eta)}(-1)^{a+b'}(-1)^{a+1'}\\sign(\\gamma,\\delta) (1^\\dual_\\delta \\wedge \\nu^\\dual_{\\gamma'\\\/\\delta}) \\otimes x = 0 \\\\\n\\Longleftrightarrow\\quad& \\ssum_{\\gamma'} \\nvect^\\dual_{\\gamma'\\\/\\delta} = 0.\n\\end{align*}\nNotice that the sum has no term if $\\maxsed(\\delta)\\neq\\maxsed(\\eta)$. Otherwise, the sum is over facets of $\\delta$ of same sedentarity and such that $\\gamma_\\infty=\\delta_\\infty$. Thus, every linear forms of the sum is zero on $\\delta_\\infty$.\nLet $u$ and $u'$ be two vertices of $\\delta_\\f$. For any $\\gamma'$,\n\\[ \\nvect^\\dual_{\\gamma'\\\/\\delta}(u'-u) = \\begin{cases}\n  1 & \\text{if $u'\\in\\gamma'$ and $u\\not\\in\\gamma'$,} \\\\\n  -1 & \\text{if $u\\in\\gamma'$ and $u'\\not\\in\\gamma'$} \\\\\n  0 & \\text{otherwise.}\n\\end{cases} \\]\nThe first condition can be rewritten as \\enquote{if $\\gamma'=\\delta-u$} and the second condition as \\enquote{if $\\gamma'=\\delta-u'$}. Thus, the first condition and the second condition each occurs exactly once. Hence, the sum of form applied to $u-u'$ is zero. To conclude, notice that $\\TT\\delta$ is spanned by the union of the edges of $\\delta_\\f$ and of $\\delta_\\infty$.\n\n\\medskip\n\n\\item \\label{com:sac} If $\\maxsed(\\delta)=\\maxsed(\\zeta)=\\maxsed(\\eta)$, we have to show that $(\\d\\d^\\i+\\d^\\i\\d)\\bigl[\\AA_\\ST^{a,b'}[\\zeta]\\to\\AA^{a+1,0,b'}[\\delta,\\eta]\\bigr](x) = 0$. This is equivalent to\n\\begin{align*}\n\\Longleftrightarrow\\quad& (-1)^{a+1+b'}\\sign(\\zeta,\\eta) 1^\\dual_\\delta \\otimes \\i^*_{\\zeta\\\/\\eta}(x)\n  \\\\ &\\qquad + \\hspace{-2em}\\sum_{\\gamma'\\ssubface\\delta \\\\ \\maxsed(\\gamma')=\\maxsed(\\delta)}\\hspace{-2em} \\sign(\\zeta,\\eta)(-1)^{a+b'} \\nvect^\\dual_{\\gamma'\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta\/\\gamma'})) \\p^*_{\\gamma'\\\/\\delta}(1^\\dual_{\\gamma'}) \\otimes \\i^*_{\\zeta\\\/\\eta}(x) = 0 \\\\\n\\Longleftrightarrow\\quad& 1 = \\ssum_{\\gamma'} \\nvect^\\dual_{\\gamma'\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta\/\\gamma'})).\n\\end{align*}\nLet $v$ be any vertex of $\\delta_\\f$. We have already seen in \\eqref{com:saa} that\n\\[ \\ssum_{\\gamma'} \\nvect^\\dual_{\\gamma'\\\/\\delta}=0. \\]\nThus,\n\\[ \\ssum_{\\gamma'} \\nvect^\\dual_{\\gamma'\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta\/\\gamma'})) = \\ssum_{\\gamma'} \\nvect^\\dual_{\\gamma'\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta\/\\gamma'}-\\nvect_{\\eta-\\zeta\/v})). \\]\nFor each $\\gamma'$, set $u_{\\gamma'}:=\\nvect_{\\eta-\\zeta\/\\gamma'}-\\nvect_{\\eta-\\zeta\/v}$. Then $u_{\\gamma'}$. is a vector going from some point of $\\TT\\gamma'$ to $v$. Thus, if $v\\in\\gamma'$, then $u_{\\gamma'}\\in\\TT\\gamma'$ and\n\\[ \\nvect^\\dual_{\\gamma'\\\/\\delta}(\\p_\\delta(u_{\\gamma'})) = 0. \\]\nOtherwise, i.e., for $\\gamma'=\\delta-v$, up to an element of $\\TT\\gamma'$, $u_{\\delta-v}$ will be equal to $\\nu_{\\delta\/\\gamma}$, and\n\\[ \\nvect^\\dual_{\\gamma'\\\/\\delta}(\\p_\\delta(u_{\\delta-v})) = 1. \\]\nThus, the sum equals $1$.\n\n\\medskip\n\n\\item \\label{com:a_easy} We have to show that $\\d_\\Dnop^2=\\d_\\Dnop\\d'_\\Dnop+\\d'_\\Dnop\\d_\\Dnop=\\d_\\Dnop\\d^\\pi_\\Dnop+\\d^\\pi_\\Dnop\\d_\\Dnop=\\d_\\Dnop'\\d_\\Dnop^\\pi+\\d_\\Dnop^\\pi\\d_\\Dnop'=(\\d^\\pi_\\Dnop)^2=0$. This follows directly from the definitions, the property of $\\sign(\\cdot,\\cdot)$, \\eqref{com:eqn:nu*}, \\eqref{com:eqn:zeta_pi} and \\eqref{com:eqn:pi_nu*}.\n\n\\medskip\n\n\\item \\label{com:abb} We need to show that $\\d'^2_\\Dnop=0$; see the proof that $\\gys^2=0$ in Section \\ref{sec:proofapp1}.\n\n\\medskip\n\n\\item \\label{com:aac} This is the property $\\d\\d^\\i+\\d^\\i\\d\\bigl[\\AA_\\Dnop^{a,b,b'}[\\gamma,\\eta] \\to \\AA_\\Dnop^{a,b,b'}[\\chi,\\mu]\\bigr](\\alpha \\otimes x) = 0$, which is equivalent to\n\\begin{align*}\n\\Longleftrightarrow\\quad& \\sum_{\\ddelta\\in\\{\\delta,\\delta'\\}}(-1)^{a+1+b'}\\sign(\\ddelta,\\chi)\\sign(\\eta,\\mu)\\nvect^\\dual_{\\gamma\\\/\\ddelta}(\\p_{\\ddelta}(\\nvect_{\\mu-\\eta\/\\gamma}))\\p^*_{\\gamma\\\/\\ddelta}(\\alpha)\\wedge\\nu^\\dual_{\\ddelta\\\/\\chi} \\otimes \\i^*_{\\eta\\\/\\mu}(x)\n  \\\\&\\qquad + (-1)^{a+b'}\\sign(\\eta,\\mu)\\nvect^\\dual_{\\ddelta\\\/\\chi}(\\p_\\chi(\\nvect_{\\mu-\\eta\/\\ddelta}))\\sign(\\gamma,\\ddelta) \\p^*_{\\ddelta\\\/\\chi}(\\alpha \\wedge \\nu^\\dual_{\\gamma\\\/\\ddelta}) \\otimes \\i^*_{\\eta\\\/\\mu}(x) = 0 \\\\\n\\Longleftrightarrow\\quad&\n  \\ssum_\\ddelta -\\sign(\\ddelta,\\chi)\\nvect^\\dual_{\\gamma\\\/\\ddelta}(\\p_{\\ddelta}(\\nvect_{\\mu-\\eta\/\\gamma})) (\\alpha\\wedge\\nu^\\dual_{\\gamma\\\/\\ddelta}\\vee\\nu_{\\ddelta\/\\gamma})\\wedge\\nu^\\dual_{\\ddelta\\\/\\chi}\n    & \\eqref{com:eqn:p_nu*_nu}\n  \\\\&\\qquad + \\sign(\\gamma,\\ddelta)\\nvect^\\dual_{\\ddelta\\\/\\chi}(\\p_\\chi(\\nvect_{\\mu-\\eta\/\\ddelta})) \\alpha\\wedge\\nu^\\dual_{\\gamma\\\/\\ddelta}\\wedge\\nu^\\dual_{\\ddelta\\\/\\chi}\\vee\\nu_{\\chi\/\\ddelta} = 0 \\\\\n\\Longleftrightarrow\\quad&\n  \\alpha\\wedge\\nu^\\dual_{\\gamma\\\/\\chi}\\Bigl( \\ssum_\\ddelta \\sign(\\ddelta,\\chi)\\nvect^\\dual_{\\gamma\\\/\\ddelta}(\\p_{\\ddelta}(\\nvect_{\\mu-\\eta\/\\gamma}))\\nu_{\\ddelta\/\\gamma}  +  \\sign(\\gamma,\\ddelta)\\nvect^\\dual_{\\ddelta\\\/\\chi}(\\p_\\chi(\\nvect_{\\mu-\\eta\/\\ddelta}))\\nu_{\\chi\/\\ddelta} \\Bigr) = 0\n    & \\eqref{com:eqn:wedge_vee},\\eqref{com:eqn:nu*} \\\\\n\\Longleftarrow\\quad&\n  \\ssum_\\ddelta \\sign(\\ddelta,\\chi)\\sign(\\gamma,\\ddelta)\\p_\\ddelta(\\nvect_{\\mu-\\eta\/\\gamma}) + \\sign(\\gamma,\\ddelta)\\sign(\\ddelta,\\chi)\\p_\\chi(\\nvect_{\\mu-\\eta\/\\ddelta}) = 0\n    & \\eqref{com:eqn:nu*_nu} \\\\\n\\Longleftrightarrow\\quad&\n  \\p_\\delta(\\nvect_{\\mu-\\eta\/\\gamma}) + \\p_\\chi(\\nvect_{\\mu-\\eta\/\\delta}) = \\p_{\\delta'}(\\nvect_{\\mu-\\eta\/\\gamma}) + \\p_\\chi(\\nvect_{\\mu-\\eta\/\\delta'}) \\\\\n\\Longleftrightarrow\\quad&\n  \\p_\\chi \\p_\\delta(\\nvect_{\\mu-\\eta\/\\gamma}) + \\p_\\chi\\cp_\\delta(\\nvect_{\\mu-\\eta\/\\gamma}) = \\p_\\chi \\p_{\\delta'}(\\nvect_{\\mu-\\eta\/\\gamma}) + \\p_\\chi\\cp_{\\delta'}(\\nvect_{\\mu-\\eta\/\\gamma}) \\\\\n\\Longleftrightarrow\\quad& \\p_\\chi(\\nvect_{\\mu-\\eta\/\\gamma}) = \\p_\\chi(\\nvect_{\\mu-\\eta\/\\gamma}),\n\\end{align*}\nwhich is obvious.\n\n\\vspace{.7cm}\n\n\nFor cases \\eqref{com:abc}, \\eqref{com:abc'} and \\eqref{com:abc''}, recall our naming convention of faces: we have $\\gamma \\ssubface \\eta, \\eta'$. We have to prove that\n\\begin{gather*}\n(\\d'\\d^\\i+\\d^\\i\\d')\\bigl[\\AA^{a,b,b'}_\\Dnop[\\gamma,\\eta'] \\to \\AA^{a+1,b,b'+1}_\\Dnop[\\delta,\\eta]\\bigr], \\textrm{ and } \\\\\n(\\d'\\d^\\i+\\d^\\i\\d')\\bigl[\\AA^{a,b,b'}_\\Dnop[\\gamma,\\eta] \\to \\AA^{a+1,b,b'+1}_\\Dnop[\\delta,\\eta]\\bigr](\\alpha \\otimes x) \\text{ equals 0}.\n\\end{gather*}\nFor the first equation, we consider two cases depending on whether there exists $\\mu$ such that $\\eta,\\eta' \\ssubface \\mu$ or not.\n\n\\medskip\n\n\\item \\label{com:abc} If $\\mu$ exists, then we have\n\\begin{align*}\n&\\hspace{-2cm}(\\d'\\d^\\i+\\d^\\i\\d')\\bigl[\\AA^{a,b,b'}_\\Dnop[\\gamma,\\eta'] \\to \\AA^{a+1,b,b'+1}_\\Dnop[\\delta,\\eta]\\bigr](\\alpha \\otimes x)=0\\\\\n\\Longleftrightarrow\\quad&\n  \\sign(\\eta,\\mu)\\sign(\\eta',\\mu)\\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\mu-\\eta'\/\\gamma}))\\p^*_{\\gamma\\\/\\delta}(\\alpha) \\otimes \\gys^{\\mu\/\\eta}\\i^*_{\\eta'\\\/\\mu}(x)\n  \\\\&\\qquad + \\sign(\\zeta,\\eta)\\sign(\\zeta,\\eta') \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta\/\\gamma})) \\p^*_{\\gamma\\\/\\delta}(\\alpha) \\otimes \\i^*_{\\zeta\\\/\\eta}\\gys^{\\eta'\/\\zeta}(x) \\\\\n\\Longleftrightarrow\\quad&\n  \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\mu-\\eta'\/\\gamma}))\n  \\\\&\\qquad = \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta\/\\gamma})) \\\\\n\\Longleftrightarrow\\quad& \\mu-\\eta'=\\eta-\\zeta,\n\\end{align*}\nwhich is obviously the case.\n\n\\medskip\n\n\\item \\label{com:abc'} If $\\mu$ does not exist, then $(\\d'\\d^\\i+\\d^\\i\\d')\\bigl[\\AA^{a,b,b'}_\\Dnop[\\gamma,\\eta'] \\to \\AA^{a+1,b,b'+1}_\\Dnop[\\delta,\\eta]\\bigr](\\alpha \\otimes x)=0$. We only have the second term from the case \\eqref{com:abc}. However, here, $\\i^*_{\\zeta\\\/\\eta}\\gys^{\\eta'\/\\zeta}(x)=0$ since $\\eta_\\infty^\\zeta$ and $\\eta'^\\zeta_\\infty$ are not comparable.\n\n\\medskip\n\n\\item \\label{com:abc''} It remains to prove $(\\d'\\d^\\i+\\d^\\i\\d')\\bigl[\\AA^{a,b,b'}_\\Dnop[\\gamma,\\eta] \\to \\AA^{a+1,b,b'+1}_\\Dnop[\\delta,\\eta]\\bigr](\\alpha \\otimes x)=0.$ This is equivalent to\n\\begin{align*}\n\\Longleftrightarrow\\quad&\n  \\sum_{\\mu' \\ssupface \\eta \\\\ \\sed(\\mu')=\\sed(\\eta)}\\sign(\\eta,\\mu')\\sign(\\eta,\\mu')\\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\mu'-\\eta\/\\gamma}))\\p^*_{\\gamma\\\/\\delta}(\\alpha) \\otimes \\gys^{\\mu'\/\\eta}\\i^*_{\\eta\\\/\\mu'}(x)\n  \\\\&\\qquad + \\sum_{\\zeta' \\ssubface \\eta \\\\ \\zeta' \\succ \\gamma \\\\ \\sed(\\zeta')=\\sed(\\eta)} \\sign(\\zeta',\\eta)\\sign(\\zeta',\\eta) \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta'\/\\gamma})) \\p^*_{\\gamma\\\/\\delta}(\\alpha) \\otimes \\i^*_{\\zeta'\\\/\\eta}\\gys^{\\eta\/\\zeta'}(x) = 0 \\\\\n\\Longleftrightarrow\\quad&\n  \\ssum_{\\mu'}\\p_\\delta^*(\\nvect^\\dual_{\\gamma\\\/\\delta})(\\nvect_{\\mu'-\\eta\/\\gamma})\\gys^{\\mu'\/\\eta}\\i^*_{\\eta\\\/\\mu'}(1^\\eta)\n  + \\ssum_{\\zeta'} \\p_\\delta^*(\\nvect^\\dual_{\\gamma\\\/\\delta})(\\nvect_{\\eta-\\zeta'\/\\gamma})\\i^*_{\\zeta'\\\/\\eta}\\gys^{\\eta\/\\zeta'}(1^\\eta) = 0 \\\\\n\\Longleftrightarrow\\quad&\n  \\ssum_{\\mu'}(\\p^\\gamma_\\delta)^*(\\e^\\dual_{\\gamma\\\/\\delta})(\\e_{\\mu'-\\eta\/\\gamma})\\i^*_{\\gamma\\\/\\eta}(x_{\\mu'-\\eta\/\\gamma})\n  + \\ssum_{\\zeta'} (\\p^\\gamma_\\delta)^*(\\e^\\dual_{\\gamma\\\/\\delta})(\\e_{\\eta-\\zeta'\/\\gamma})\\i^*_{\\gamma\\\/\\eta}(x_{\\eta-\\zeta'\/\\gamma}) = 0.\n\\end{align*}\n\nNotice now that rays of the form $\\mu'^\\gamma-\\eta^\\gamma$ are exactly the rays of $\\Sigma^\\gamma$ in the link of $\\eta^\\gamma$, and the rays of the form $\\eta^\\gamma-\\zeta'^\\gamma$ are exactly the rays of $\\eta^\\gamma$. Moreover, if $\\varrho$ is a ray which is not in the link of $\\eta^\\gamma$ nor in $\\eta^\\gamma$, then $\\i^*_{\\gamma\\\/\\eta}(x_\\varrho)=0$. Thus, the above sum equals\n\\begin{align*}\n\\i^*_{\\gamma\\\/\\eta}\\Bigl(\\sum_{\\varrho\\in\\Sigma^\\gamma \\\\ \\dims\\varrho = 1} (\\p^\\gamma_\\delta)^*(\\e^\\dual_{\\gamma\\\/\\delta})(\\e_\\varrho)x_\\varrho\\Bigr),\n\\end{align*}\nwhich is clearly zero.\n\n\\medskip\n\n\\item \\label{com:acc} We have to show that $(\\d^\\i)^2\\bigl[\\AA_\\Dnop^{a,b,b'}[\\gamma,\\zeta] \\to \\AA_\\Dnop^{a+2,b,b'}[\\chi,\\mu]\\bigr](\\alpha \\otimes x)=0$. This is equivalent to\n\\begin{align*}\n\\Longleftrightarrow\\quad&\n  \\sum_{\\eeta \\in \\{\\eta,\\eta'\\}}\\sum_{\\ddelta \\in \\{\\delta,\\delta'\\} \\\\ \\ddelta \\prec \\eeta} \\sign(\\eeta,\\mu)\\nvect^\\dual_{\\ddelta\\\/\\chi}(\\p_\\chi(\\nvect_{\\mu-\\eeta\/\\ddelta})) \\sign(\\zeta,\\eeta)\\nvect^\\dual_{\\gamma\\\/\\ddelta}(\\p_\\ddelta(\\nvect_{\\eeta-\\zeta\/\\gamma})) \\p^*_{\\ddelta\\\/\\chi}\\p^*_{\\gamma\\\/\\ddelta}(\\alpha) \\otimes \\i^*_{\\eeta\\\/\\mu}\\i^*_{\\zeta\\\/\\eeta}(x)=0 \\\\\n\\Longleftrightarrow\\quad&\n  \\ssum_{\\eeta,\\ddelta} \\sign(\\eeta,\\mu)\\sign(\\zeta,\\eeta)\\sign(\\ddelta,\\chi)\\sign(\\gamma,\\ddelta) \\nu^\\dual_{\\ddelta\\\/\\chi}(\\p_\\chi\\cp_\\ddelta(\\nvect_{\\mu-\\eeta\/\\gamma})) \\nu^\\dual_{\\gamma\\\/\\ddelta}(\\p_\\chi \\p_\\ddelta(\\nvect_{\\eeta-\\zeta\/\\gamma})) =0 \\\\\n\\Longleftrightarrow\\quad&\n  \\ssum_{\\eeta,\\ddelta} \\sign(\\eeta,\\mu)\\sign(\\zeta,\\eeta)\\sign(\\ddelta,\\chi)\\sign(\\gamma,\\ddelta) \\nu^\\dual_{\\gamma\\\/\\chi}(\\cp_\\ddelta \\p_\\chi(\\nvect_{\\mu-\\eeta\/\\gamma}) \\wedge \\p_\\ddelta \\p_\\chi(\\nvect_{\\eeta-\\zeta\/\\gamma})) =0.\n\\end{align*}\n\nNotice that, if $\\delta'\\not\\prec\\eta$, then $\\mu-\\eta=\\delta'-\\gamma$ and $\\cp_{\\delta'}\\p_\\chi(\\nvect_{\\mu-\\eta\/\\gamma})=0$. Using the same argument for $\\delta$ and $\\eta'$, this shows that we can now remove the condition $\\ddelta\\prec\\eeta$ in the above sum.\n\n\\smallskip\n\\par Set $u=\\p_\\chi(\\nvect_{\\mu-\\eta'\/\\gamma})=\\p_\\chi(\\nvect_{\\eta-\\zeta\/\\gamma})$ and $u'=\\p_\\chi(\\nvect_{\\mu-\\eta\/\\gamma})=\\p_\\chi(\\nvect_{\\eta'-\\zeta\/\\gamma})$. The above sum equals\n\\begin{align*}\n\\sign(\\eta,\\mu)\\sign(\\zeta,\\eta)\\sign(\\delta,\\chi)\\sign(\\gamma,\\delta)\\nu^\\dual_{\\gamma\\\/\\chi}\\Bigl(\\cp_\\delta(u')\\wedge \\p_\\delta(u) - \\cp_\\delta(u)\\wedge \\p_\\delta(u') - \\cp_{\\delta'}(u')\\wedge \\p_{\\delta'}(u) + \\cp_{\\delta'}(u)\\wedge \\p_{\\delta'}(u')\\Bigr).\n\\end{align*}\nThen we have\n\\begin{align*}\n&\\cp_\\delta(u')\\wedge \\p_\\delta(u) - \\cp_\\delta(u)\\wedge \\p_\\delta(u') \\\\\n&\\qquad= \\bigl(\\cp_\\delta(u')+\\p_\\delta(u')\\bigr)\\wedge\\bigl(\\cp_\\delta(u)+\\p_\\delta(u)\\bigr) - \\cp_\\delta(u') \\wedge \\cp_\\delta(u) - \\p_\\delta(u')\\wedge \\p_\\delta(u) \\\\\n&\\qquad= u' \\wedge u - 0 - \\pi_\\delta(u') \\wedge \\pi_\\delta(u).\n\\end{align*}\nNotice that $\\nu^\\dual_{\\gamma\\\/\\chi}(\\v)=0$ if $\\v\\in\\bigwedge^2\\TT\\delta$. Thus, up to a sign, the above sum equals\n\\[ \\nu^\\dual_{\\gamma\\\/\\chi}(u' \\wedge u + u \\wedge u') = 0. \\]\n\n\\medskip\n\n\\item \\label{com:acd} Finally, we have to show that $(\\d^\\i\\d^\\pi+\\d^\\pi\\d^\\i)\\bigl[\\AA_\\Dnop^{a,b,b'}[\\gamma,\\zeta] \\to \\AA_\\Dnop^{a+2,b,b'}[\\chi,\\mu]\\bigr](\\alpha \\otimes x)=0$ with $\\sed(\\mu)=\\sed(\\chi)=\\sed(\\eta')=\\sed(\\delta')\\ssubface\\sed(\\eta)=\\sed(\\delta)=\\sed(\\zeta)=\\sed(\\gamma)$. This is equivalent to\n\\begin{align*}\n&\n  \\sign(\\eta',\\mu)\\nvect^\\dual_{\\delta'\\\/\\chi}(\\p_\\chi(\\nvect_{\\mu-\\eta'\/\\delta'}))\\sign(\\zeta,\\eta') \\p^*_{\\delta'\\\/\\chi}\\pi^{\\sed*}_{\\gamma\/\\delta'}(\\alpha) \\otimes \\i^*_{\\eta'\\\/\\mu}\\id^{\\zeta\\\/\\eta'}(x)\n  \\\\*&\\qquad + \\sign(\\eta,\\mu)\\sign(\\zeta,\\eta) \\nvect^\\dual_{\\gamma\\\/\\delta}(\\p_\\delta(\\nvect_{\\eta-\\zeta\/\\gamma})) \\pi^{\\sed *}_{\\delta\\\/\\chi}\\p^*_{\\gamma\\\/\\delta}(\\alpha) \\otimes \\id^{\\eta\\\/\\mu}\\i^*_{\\zeta\\\/\\eta}(x) = 0,\n\\end{align*}\nwhich is implied by\n\\begin{align*}\n&\n  \\e^\\dual_{\\delta'\\\/\\chi}(\\p^{\\delta'}_\\chi(\\e_{\\mu-\\eta'\/\\delta'})) = \\e^\\dual_{\\gamma\\\/\\delta}(\\p^{\\gamma}_\\delta(\\e_{\\eta-\\zeta\/\\gamma}))\\text{ and }  \\pi^{\\sed}_{\\gamma\\\/\\delta'}\\p_{\\delta'\\\/\\chi} = \\p_{\\gamma\\\/\\delta}\\pi^{\\sed}_{\\delta\\\/\\chi}.\n\\end{align*}\n\n\\end{enumerate}\n\n\\medskip\n\n\n\n\n\n\n\\section{Tropical monodromy}\n\nIn this section, we prove Theorem \\ref{thm:tropical_monodromy} that describes the simplicial tropical monodromy operator and claims that it corresponds to the monodromy operator on Steenbrink. We use the notations and conventions of Section \\ref{sec:technicalities}. Recall that there is one triple complex $\\AA^{\\bul,\\bul,\\bul}$ for each even row $\\ST^{\\bul,2p}_1$ of the steenbrink spectral sequence. To distinguish these spectral sequences, we precise the corresponding $p$ as follows: $\\AAp{p}^{\\bul,\\bul,\\bul}$.\n\n\\medskip\n\nFor each face $\\delta\\in X_\\f$, let $o_\\delta$ be a point of $\\Tan\\delta\\subseteq N_\\mathbb{R}$, for instance the centroid of $\\delta$. For other faces of $\\delta$ of sedentarity $\\conezero$, set $o_\\delta:=o_{\\delta_\\f}$. For any faces $\\delta$ of any sedentarity $\\sigma$, set $o_\\delta=\\pi^\\sigma(o_\\eta)$, where $\\eta$ is the only face of sedentarity $\\conezero$ such that $\\delta=\\eta_\\infty^\\sigma$. We define $v_{\\delta\\\/\\eta}:=o_\\eta-o_\\delta$.\n\n\\medskip\n\nLet $V$ and $W$ be any vector spaces such that $\\TT\\delta\\subseteq V$ and $k$ be an integer. We define the morphism $\\~N_{\\gamma\\\/\\delta}\\colon \\bigwedge^kV^\\dual\\otimes W\\to\\bigwedge^{k-1}V^\\dual\\otimes W$ by\n\\[ \\~N_{\\gamma\\\/\\delta} = (\\,\\cdot \\vee v_{\\gamma\\\/\\delta}) \\otimes \\id\\!. \\]\n\nWe define a map between triple complexes $N\\colon\\AAp{p}^{\\bul,\\bul,\\bul} \\to \\AAp{p-1}^{\\bul,\\bul,\\bul}$ as follows. Assume $\\gamma$ is a face of dimension $a+b$ and $\\eta$ of dimension $a+p-b'$, with $b'\\leq p$, and such that $\\dims{\\eta_\\infty}\\leq a$. Then,\n\\begin{align*}\nN\\Bigl[\\,\\AAp{p}_\\trop^{a,b}[\\gamma] \\to \\AAp{p-1}^{\\bul,\\bul}_\\trop[\\delta] \\,\\Bigr]\n  &= \\~N_{\\gamma\\\/\\delta}(\\d_\\trop+\\dfrak_\\trop), \\\\\n  &\\\\\nN\\Bigl[\\, \\AAp{p}_\\Dnop^{a,b,b'}[\\gamma, \\eta] \\to \\AAp{p-1}_\\Dnop^{\\bul,\\bul,\\bul}[\\delta, \\eta]\\, \\Bigr]\n  &= \\~N_{\\gamma\\\/\\delta}(\\d_\\Dnop+\\dfrak_\\Dnop), \\\\\n  &\\\\\nN\\Bigl[\\,\\AAp{p}_\\ST^{a,b'}[\\eta] \\to \\AAp{p-1}_\\ST^{a+1,b'}[\\eta]\\,\\Bigr]\n  &= \\begin{cases}\n    \\id & \\text{if $b'\\leq p-1$,} \\\\\n    0 & \\text{otherwise.}\n  \\end{cases}\n\\end{align*}\n\n\\begin{prop} \\label{prop:com:N_commutes}\nThe map $N$ commutes with the differentials of $\\AAp{p}^{\\bul,\\bul,\\bul}$ and $\\AAp{p-1}^{\\bul,\\bul,\\bul}$.\n\\end{prop}\n\n\\begin{proof}\nThe fact that $N\\d^\\pi+\\d^\\pi N=0$ is easy to check.\n\n\\medskip\n\nLet us prove that $(N\\d-\\d N)\\bigl[\\AAp{p}_{\\trop}^{a,b}[\\gamma] \\to \\AAp{p-1}_{\\trop}^{a+1,b+1}[\\chi]\\bigr]=0$. The key point is that\n\\[ (\\d\\~N_{\\gamma\\\/\\delta} + \\~N_{\\gamma\\\/\\delta}\\d)\\bigl [\\AAp{p}_{\\trop}[\\eta] \\to *\\bigr] = 0, \\]\nwhich can easily been checked. We simplify the notations by writing $[ \\gamma \\to \\chi]$ instead of $\\bigl[\\AAp{p}_{\\trop}^{a,b}[\\gamma] \\to \\AAp{p-1}_{\\trop}^{a+1,b+1}[\\chi]\\bigr]$, or when the corresponding domain and codomain are clear.\n\n\\medskip\n\n Then we get\n\\begin{align*}\n(N\\d-\\d N)[\\gamma \\to \\chi]\n  &= \\sum_{\\ddelta\\in\\{\\delta,\\delta'\\}} N[\\ddelta \\to \\chi]\\d[\\gamma \\to \\ddelta]-\\d[\\ddelta \\to \\chi]N[\\gamma \\to \\ddelta] \\\\\n  &= \\sum_{\\ddelta\\in\\{\\delta,\\delta'\\}} (\\~N_{\\ddelta\\\/\\chi}+\\~N_{\\gamma\\\/\\ddelta})\\d[\\ddelta \\to \\chi]\\d[\\gamma \\to \\ddelta].\n\\end{align*}\nFrom $\\d^2=0$, we deduce that $\\d[\\delta' \\to \\chi]\\d[\\gamma \\to \\delta']=-\\d[\\delta \\to \\chi]\\d[\\gamma \\to \\delta]$. Thus,\n\\begin{align*}\n(N\\d-\\d N)[\\gamma \\to \\chi]\n  &= (\\~N_{\\delta\\\/\\chi}+\\~N_{\\gamma\\\/\\delta}-\\~N_{\\delta'\\\/\\chi}-\\~N_{\\gamma\\\/\\delta'})\\d[\\delta\\to\\chi]\\d[\\gamma\\to\\delta].\n\\end{align*}\nBy definition of $\\~N$, to prove this is zero, it suffices to check that\n\\[ v_{\\delta\\\/\\chi}+v_{\\gamma\\\/\\delta}-v_{\\delta'\\\/\\chi}-v_{\\gamma\\\/\\delta'}=0. \\]\nBut this vector is by definition\n\\[ (o_\\chi-o_\\delta)+(o_\\delta-o_\\gamma)-(o_\\chi-o_{\\delta'})-(o_{\\delta'}-o_\\gamma), \\]\nwhich is clearly zero.\n\n\\medskip\n\nOne can prove with a similar argument that $N$ and $\\d+\\d'+\\dfrak$ commute on the tropical part and also on the $\\Dnop$ part.\n\n\\medskip\n\nFor the Steenbrink part, Proposition \\ref{prop:N_commutes} proves that $N$ commutes with $\\d'_\\ST$ and $\\d^\\i_\\ST$ on $X_\\f$. The proposition can easily been extended to all the faces, and the commutativity with $\\d^\\pi_\\ST$ is also easy to check. Thus, it only remains to see the commutativity with $\\d_\\ST$, i.e., with the inclusion $\\AA_\\ST^{\\bul,\\bul} \\hookrightarrow \\AA_\\Dnop^{\\bul,0,\\bul}$.\n\n\\medskip\n\nWe have\n\\begin{align*}\n&\\hspace{-2em}(\\d N-N\\d)\\bigl[\\AAp{p}_\\ST^{a,b'}[\\eta] \\to \\AAp{p-1}_\\Dnop^{a,b',0}[\\eta,\\delta]\\bigr](x) \\\\\n  &= (-1)^{a+1+b'} 1^\\dual_\\delta \\otimes x\n    - \\hspace{-2em}\\sum_{\\gamma' \\ssubface \\delta \\\\ \\maxsed(\\gamma')=\\maxsed(\\delta)}\\hspace{-2em} (-1)^{a+b'}\\sign(\\gamma',\\delta) v_{\\gamma'\\\/\\delta} \\vee (1^\\dual_{\\gamma'}\\wedge\\nu^\\dual_{\\gamma'\\\/\\delta}) \\otimes x \\\\\n  &= \\bigl( -1 + \\ssum_{\\gamma'}\\nvect^\\dual_{\\gamma'\\\/\\delta}(v_{\\gamma'\\\/\\delta}) \\bigr) 1^\\dual_\\delta \\otimes x.\n\\end{align*}\nThe terms of the sum are the normalized barycentric coordinates of $o_\\delta$ in $\\delta_\\f$. Hence, the terms sum up to one. This concludes the proof of the commutativity of $N$ with the differentials.\n\\end{proof}\n}\n\n\n\n\n\\newpage\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGiven a graph class $\\mathcal{C}$, a graph $G=(V,E)$ is called a \\emph{probe $\\mathcal{C}$ graph} if there exists an independent set $\\N\\subseteq V$ (of \\emph{non-probes}) and a set $E'\\subseteq\\binom{\\N}{2}$ such that the graph $G'=(V,E\\cup E')$ is in\nthe class $\\mathcal{C}$, where $\\binom{\\N}{2}$ stands for the set of all 2-element subsets of $\\N$.\nA graph $G=(V,E)$ with a \\emph{given} independent set $\\N\\subseteq V$ is said to be a \\emph{partitioned probe $\\mathcal{C}$ graph} if there exists a set $E'\\subseteq\\binom{\\N}{2}$ such that the graph $G'=(V,E\\cup E')$ is in the class $\\mathcal{C}$. In both cases, $G'$ is called a $\\mathcal{C}$ {\\em embedding} of $G$. Thus, a graph is a (partitioned) probe $\\mathcal{C}$ graph if and only if it admits a $\\mathcal{C}$ embedding.\n\nRecognizing partitioned probe $\\mathcal{C}$ graphs is a special case of the $\\mathcal{C}$-\\textsc{graph sandwich} problem (cf.~\\cite{GolKapSha95}). More precisely, given two graphs $G_i=(V, E_i)$, $i=1,2$, on the same vertex $V$ such that $E_1\\subseteq E_2$, the $\\mathcal{C}$-\\textsc{graph sandwich} problem asks for the existence of a graph $G=(V,E)$ such that $E_1\\subseteq E\\subseteq E_2$ and $G$ is in $\\mathcal{C}$. Recognizing partitioned probe $\\mathcal{C}$ graphs with a given independent set $\\N$ is a special case of the $\\mathcal{C}$-\\textsc{graph sandwich} problem, where $E_2\\setminus E_1=\\binom{\\N}{2}$. Both concepts stem from computational biology; see, {\\em e.g.\\\/},~\\cite{GolKapSha94,GolKapSha95,Zhang,Zhang-etal}.\n\nProbe graphs have been investigated for various graph classes; see~\\cite{ChaChaKloPen} for more information.\n\nRecently, the concept of probe graphs has been generalized in~\\cite{ChaHunKloPen}.\nA graph $G$ is said to be a \\emph{$k$-probe $\\mathcal{C}$ graph} if there exist independent sets $\\N_1, \\ldots, \\N_k$ in $G$ such that there exists a graph $G'\\in \\mathcal{C}$ (an \\emph{embedding} of $G$) such that for every edge $xy$ in $G'$ which is not an edge of $G$ there\nexists an $i$ with $x, y \\in \\N_i$. In the case $k = 1$, $G$ is a \\emph{probe $\\mathcal{C}$ graph}.\n\nWe refer to the partitioned case of the problem when a collection of independent sets\n$\\N_i$, $i = 1,\\ldots, k$, is a part of the input; otherwise, it is an unpartitioned case.\nFor historical reasons we call the set of vertices $\\Probe = V \\setminus \\bigcup^k_{i=1} \\N_i$ the set of \\emph{probes} and the vertices of $\\bigcup^k_{i=1} \\N_i$ the set of\n\\emph{non-probes}.\n\nIn~\\cite{ChaHunKloPen}, $k$-probe complete graphs and $k$-probe block graphs have been investigated. The authors proved that, for fixed $k$, $k$-probe complete graphs can be characterized by finitely many forbidden induced subgraphs, their proof is however not constructive. They also showed, implicitly, that $k$-probe complete graphs and $k$-probe block graphs can be recognized in cubic time. The case $k=1$, {\\em e.g.}, probe complete graphs and probe block graphs, has been discussed in depth in~\\cite{LePeng12}.\n\nIn this paper, we study $2$-probe complete graphs and $2$-probe block graphs in more details. Our main results are:\n\\begin{itemize}\n \\item A characterization of partitioned $2$-probe block graphs in terms of certain ``enhanced graph'' (Theorem~\\ref{thm:p2-probeblock}), stating that $G$ is a partitioned $2$-probe block graph if and only if the enhanced graph $G^*$ is a block graph.\n \\item Forbidden induced subgraph characterizations of unpartitioned $2$-probe block graphs (Theorem~\\ref{thm:2-probeblock}).\n \\item Linear time recognition for $2$-probe block graphs, in both partitioned and unpartitioned cases.\n\\end{itemize}\nThe first result is of great interest because the enhanced graph contains only necessary edges, {\\em i.e.}, new edges that must be added. In this sense, the enhanced graph is an optimal embedding of the probe graph.\nThis type of characterization is rarely possible, and our result is the first one in case $k=2$. In case of probe graphs, {\\em i.e.}, $k=1$ only few are known: In~\\cite{BayLedeR} it is shown that a graph is a partitioned probe threshold graph, respectively, a partitioned probe trivially perfect graph if and only if a certain enhanced graph is a threshold graph, respectively, a trivially perfect graph. In~\\cite{Le} it is shown that a graph is a partitioned chain graph if and only if a certain enhanced graph is a chain graph, and recently, \\cite{LePeng12} (cf. Theorem~\\ref{thm:pprobeblock}) proved that a graph is a partitioned block graph if and only if a certain enhanced graph is a block graph. For some other cases, a certain enhanced graph can be defined that admits some nice properties; see~\\cite{CohGolLipSte,GolLip,Zhang}.\n\nForbidden induced subgraph characterizations are very desirable as they (or their proofs)\noften imply polynomial time for recognition, and give a lot of structural\ninformation of the graphs.\\footnote{That is why characterizing probe interval graphs by forbidden induced subgraphs is a long-standing interesting open problem; see~\\cite{MWZ}} This is the case with the second result. Based on our forbidden induced subgraph characterization, we will obtain a linear time algorithm for recognizing if a given graph is a $2$-probe block graph, improving the cubic time complexity provided previously in~\\cite{ChaHunKloPen}.\n\nThe paper is structured as follows. In Section~\\ref{sec:def}, we collect all the necessary\ndefinitions, and review results about probe complete graphs and probe block graphs. In Section~\\ref{sec:2-probecomplete}, we discuss $2$-probe complete graphs. Partitioned and unpartitioned $2$-probe block graphs will be considered in Section~\\ref{sec:p2-probeblock} and in Section~\\ref{sec:2-probeblock}, respectively. A linear time recognition algorithm of unpartitioned 2-probe block graphs is proposed in Section~\\ref{sec:reog}. We conclude the paper with some open problems in Section~\\ref{sec:conclusion}.\n\n\\section{Definitions and notion}\\label{sec:def}\nIn a graph, a set of vertices is an {\\em independent set}, respectively, a {\\em clique} if no two, respectively, every two vertices in this set are adjacent. For two graphs $G$ and $H$, we write $G+H$ for the disjoint union of $G$ and $H$, and $2G$ for $G+G$. The {\\em join} $G\\star H$ is obtained from $G+H$ by adding all possible edges $xy$ between any vertex $x$ in $G$ and any vertex $y$ in $H$. The complete graph with $n$ vertices is denoted by $K_n$. The path and cycle with $n$ vertices of length $n-1$, respectively, of length $n$, is denoted by $P_n$, respectively, $C_n$. Let $G = (V, E)$ be a graph. For a vertex $v\\in V$ we write $N(v)$ for the set of its neighbors in $G$. A \\emph{universal} vertex $v$ is one such that $N(v)\\cup\\{v\\}=V$.\nFor a subset $U\\subseteq V$ we write $G[U]$ for the subgraph of $G$ induced by $U$ and $G-U$ for the graph $G[V\\setminus U]$; for a vertex $v$ we write $G-v$ rather than $G[V\\setminus \\{v\\}]$.\n\nA (connected or not) graph is a \\emph{block graph} if each of its maximal 2-connected components, {\\em i.e.\\\/}, its blocks, is a clique. A \\emph{chordal graph} is one in which every cycle $C_\\ell$ of length $\\ell\\ge 4$ has a chord. (A chord of a cycle is an edge not belonging to the cycle but joining to vertices of the cycle.) A \\emph{diamond} is the complete graph on four vertices minus an edge. It is well-known (and easy to see) that block graphs are exactly the chordal graphs without induced diamond.\n\n\\begin{Proposition}[Folklore]\nA graph is a block graph if and only if it is a diamond-free chordal graph.\n\\end{Proposition}\n\nHere, given a graph $F$, a graph is said to be {\\em $F$-free} if it has no induced subgraph isomorphic to $F$.  For a set of graphs $\\mathcal{F}$, a graph is said to be {\\em $\\mathcal{F}$-free} if it is $F$-free for each $F\\in \\mathcal{F}$.\n\nA graph $G$ is called \\emph{distance-hereditary} if for all vertices $u,v \\in V(G)$ any induced path between $u$ and $v$ is a shortest path. A graph $G$ is called \\emph{ptolemaic} if, in any connected component of $G$, every four vertices\n satisfy the so-called \\emph{ptolemaic inequality} (cf.~\\cite{Howorka}).\n\n\\begin{Proposition}[Folklore]\\label{dh}\n\\mbox{}\n\\begin{itemize}\n\\item[\\em (i)] Ptolemaic graphs, gem-free chordal graphs, and $C_4$-free distance-hereditary graphs coincide.\n\\item[\\em (ii)] Distance-hereditary graphs are exactly the graphs without induced house, hole, domino, gem.\n\\end{itemize}\n\\end{Proposition}\n\nHere, a \\emph{house} is a $5$-cycle with exactly one chord, a \\emph{hole} is a $C_\\ell$, $\\ell\\ge 5$, a \\emph{domino} is a $6$-cycle with exactly one long chord, and a \\emph{gem} is the join $P_4\\star K_1$.\n\nAnother graph class that will be important in our discussion is the class of $P_4$-free graphs, or \\emph{cographs}. Clearly, by Proposition~\\ref{dh}, cographs are distance-hereditary, and we will often use the following well known fact.\n\n\\begin{Proposition}[Folklore]\nAny connected cograph $G$ is the join $G=G_1\\star G_2$ of two smaller cographs $G_1, G_2$.\n\\end{Proposition}\n\nFor graph classes not defined here see, for example,~\\cite{BraLeSpi,ChaChaKloPen,Golumbic}.\n\n\nA \\emph{split graph} is a graph whose vertex set can be partitioned into a clique and an independent set. It is well-known that split graphs are exactly the chordal graphs without induced $2K_2$.\n\n\\begin{Proposition}[\\cite{FolHam}]\\label{prop:split}\nA graph is a split graph if and only if it is a $2K_2$-free chordal graph.\n\\end{Proposition}\n\nA \\emph{complete split graph} is a split graph $G=(V,E)$ admitting a partition $V=Q\\cup S$ into a clique $Q$ and an  independent set $S$ such that every vertex in $Q$ is adjacent to every vertex in $S$. Such a partition is also called a \\emph{complete split partition} of a split graph. Note that if the complete split graph $G=(V,E)$ is not a clique, then $G$ has exactly one complete split partition $V=Q\\cup S$.\n\n\\begin{Proposition}[\\cite{LePeng12}]\\label{prop:completesplit}\nThe following statements are equivalent for any graph $G$.\n\\begin{itemize}\n \\item[\\em (i)] $G$ is a probe complete graph;\n \\item[\\em (ii)] $G$ is a $\\{K_2+K_1, C_4\\}$-free graph;\n \\item[\\em (iii)] $G$ is a $(K_2+K_1)$-free split graph;\n \\item[\\em (iv)] $G$ is a complete split graph.\n\\end{itemize}\n\\end{Proposition}\n\n\nGiven a graph $G=(V,E)$ together with an independent set $\\N\\subseteq V$, the \\emph{enhanced graph} $G^*=(V,E^*)$ is obtained from $G$ by adding all edges between two vertices in $\\N$ that are two vertices of an induced diamond in $G$.\n\nPartitioned probe block graphs can be characterized as follows.\n\n\\begin{Theorem}[\\cite{LePeng12}]\\label{thm:pprobeblock}\nLet $G=(V,E)$ be a graph with a partition $V= \\Probe\\cup \\N$, where $\\N$ is an independent set.\nThen the following statements are equivalent: \n\\begin{itemize}\n \\item[\\em (i)] $G=(\\Probe\\cup \\N, E)$ is a partitioned probe block graph;\n \\item[\\em (ii)] $G$ is a ptolemaic graph and satisfies the property that the two non-adjacent vertices of every induced diamond in $G$ belong to $\\N$;\n \\item[\\em (iii)] Every block $B$ of $G$ is a complete split graph with $B=(B\\cap \\Probe) \\cup (B\\cap \\N)$ a complete split partition;\n \\item[\\em (iv)] $G^*$ is a block graph.\n\\end{itemize}\n\\end{Theorem}\n\nNote that condition (ii) in Theorem~\\ref{thm:pprobeblock} above can be equivalently stated using three partitioned induced forbidden diamonds (cf.~\\cite{LePeng12}).\n\nProbe block graphs can be characterized as follows; see Fig.~\\ref{fig:F1234} for the graphs $F_1, F_2$ and $F_3$.\n\\begin{Theorem}[\\cite{LePeng12}]\\label{thm:probeblock}\nThe following statements are equivalent for any graph $G$:\n\\begin{itemize}\n \\item[\\em (i)] $G$ is a probe block graph;\n \\item[\\em (ii)] $G$ is an $\\{F_1, F_2$, $F_3\\}$-free ptolemaic graph;\n \\item[\\em (iii)] $G$ is an $\\{F_2$, $F_3\\}$-free\n                  graph in which every block is a probe complete graph.\n\\end{itemize}\n\\end{Theorem}\n\\begin{figure}[H\n\\begin{center}\n\n\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,0) {};\n\\node[vP] (2) at (0,1.5) {};\n\\node[vP] (3) at (1,2.5) {};\n\\node[vP] (4) at (2,1.5) {};\n\\node[vP] (5) at (2,0) {};\n\n\\draw (1) -- (2) -- (3) -- (4) -- (5) -- (1) -- (4) -- (2) -- (5);\n\n\\node[] (F1) at (1,0) [label=below:$F_1$] {};\n\\end{tikzpicture}\n\\qquad\\quad\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,0) {};\n\\node[vP] (6) at (3,2) {};\n\\node[vP] (7) at (4,1) {};\n\n\\draw (1) -- (2) -- (3) -- (4) -- (5) -- (7) -- (6) -- (4) -- (2);\n\\draw (1) -- (3); \\draw (4) -- (7);\n\n\\node[] (F2) at (2,0) [label=below:$F_2$] {};\n\\end{tikzpicture}\n\\qquad\\quad\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,1) {};\n\\node[vP] (6) at (4,0) {};\n\\node[vP] (7) at (4,2) {};\n\\node[vP] (8) at (5,1) {};\n\n\\draw (1) -- (2) -- (3) -- (4) -- (5) -- (6) -- (7) -- (8) -- (6);\n\\draw (1) -- (3); \\draw (2) -- (4); \\draw (5) -- (7);\n\n\\node[] (F3) at (2.5,0) [label=below:$F_3$] {};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Forbidden induced subgraphs for unpartitioned probe block graphs.}\n\\label{fig:F1234}\n\\end{figure}\n\\begin{Theorem}[\\cite{LePeng12}]\nPartitioned and unpartitioned probe block graphs can be recognized in linear time.\n\\end{Theorem}\n\nCondition (ii) in Theorem~\\ref{thm:probeblock} above can be interpreted as follows: The block structure is described by forbidding $F_1$, the gluing conditions for the blocks are given by forbidding $F_2$ and $F_3$.\n\n\\section{$2$-probe complete graphs}\\label{sec:2-probecomplete}\nFor fixed $k$, it was shown in~\\cite{ChaHunKloPen} that, by a non-constructive proof, $k$-probe complete graphs can be characterized by at most $2^{k+1}+1$ obstructions, and that $k$-probe complete graphs can be recognized in cubic time. Here we give the complete list of five obstructions for $2$-probe complete graphs, and provide a second characterization of $2$-probe complete graphs. These results imply a linear time recognition algorithm for $2$-probe complete graphs, and are important when discussing $2$-probe block graphs later.\n\nA graph $G=(V,E)$ is called a $(K,X,Y,Z)$-graph, written $G=(K,X,Y,Z)$, if $V$ can be partitioned into disjoint (possibly empty) subsets $K, X, Y, Z$ such that\n\\begin{itemize}\n \\item $K$ is the set of all universal vertices of $G$ (hence, $K$ is a clique),\n \\item $X\\cup Z$ and $Y\\cup Z$ are independent sets,\n \\item every vertex in $X$ is adjacent to every vertex in $Y$.\n\\end{itemize}\n\nNote that if $X=\\emptyset$ or $Y=\\emptyset$, or ($Z=\\emptyset$ and ($|X|\\le 1$ or $|Y|\\le 1$)), then a $(K,X,Y,Z)$-graph is a complete split graph. Note also that $G$ is a $(K,X,Y,Z)$-graph if and only if the graph obtained from $G$ by deleting all universal vertices has at most one nontrivial connected component which is a complete bipartite graph. Hence, $(K,X,Y,Z)$-graphs can be recognized in linear time.\n\n\\begin{Theorem}\\label{thm:2-probecomplete}\nThe following statements are equivalent for any graph $G$:\n\\begin{itemize}\n  \\item[\\em (i)] $G$ is a $2$-probe complete graph;\n  \\item[\\em (ii)]  $G$ is $\\{P_4, 2K_2, K_3+ K_1,(K_2+K_1)\\star 2K_1, 2K_1\\star 2K_1\\star 2K_1\\}$-free;\n  \\item[\\em (iii)] $G$ is a $(K,X,Y,Z)$-graph.\n\\end{itemize}\n\\end{Theorem}\n\\proof\n\n\\noindent\n(i) $\\Rightarrow$ (ii): By inspection, none of $P_4$, $2K_2$, $K_3+ K_1$, $(K_2+K_1)\\star 2K_1$ and $2K_1\\star 2K_1\\star 2K_1$\nis a $2$-probe complete graph.\n\n\\smallskip\\noindent\n(ii) $\\Rightarrow$ (iii): By induction. Let $G$ satisfy (ii). Suppose first, that $G$ is connected. Then, as $G$ is a cograph, $G$ is the join of two smaller graphs, say $G=G_1\\star G_2$. By induction, $G_i$ is a $(K_i,X_i,Y_i,Z_i)$-graph, $i=1,2$. If $G_1$ or $G_2$ is a clique, say $G_2$, then $G$ is a $(K_1\\cup V(G_2),X_1,Y_1,Z_1)$-graph. If both $G_1$ and $G_2$ are not cliques, then both $G_1$ and $G_2$ are $\\{K_2+K_1, C_4\\}$-free (otherwise $G$ would have an induced $(K_2+K_1)\\star 2K_1$ or $2K_1\\star 2K_1\\star 2K_1$.\nBy Proposition~\\ref{prop:completesplit}, for each $i=1,2$, $G_i=(Q_i,S_i)$ is a complete split graph. Hence $G$ is a $(K,X,Y,\\emptyset)$-graph with $K=Q_1\\cup Q_2$, $X=S_1, Y=S_2$. Suppose now that $G$ is disconnected. We may assume that $G$ is not edgeless. Then $G$ has exactly one nontrivial connected component (as $G$ is $2K_2$-free), say $H$. Note that $H$ is $K_3$-free (as $G$ is $(K_3+K_1)$-free), and hence $H$ is complete bipartite (as $H$ is a connected cograph). Thus $G$ is a $(\\emptyset,X,Y,Z)$-graph with $(X,Y)$ being the bipartition of $H$ and $Z=V(G)\\setminus V(H)$.\n\n\\smallskip\\noindent\n(iii) $\\Rightarrow$ (i): This is obvious by setting $\\N_1=X\\cup Z$, $\\N_2=Y\\cup Z$.\n\\qed\n\n\\begin{Corollary}\\label{cor:p2-probecomplete}\nLet $G=(V,E)$ be a graph with two independent sets $\\N_1,\\N_2$ and $\\Probe=V\\setminus\\big(\\N_1\\cup\\N_2\\big)$.\nThen $G=(\\Probe,\\N_1,\\N_2, E)$ is a partitioned $2$-probe complete graph if and only if $G$ is a $(K,X,Y,Z)$-graph such that\n$\\N_1=X\\cup Z$ and $\\N_2=Y\\cup Z$.\n\\end{Corollary}\n\nAs $(K,X,Y,Z)$-graphs can be recognized in linear time, we obtain:\n\\begin{Corollary}\\label{cor:recog-p2-probecomplete}\nUnpartitioned and partitioned $2$-probe complete graphs can be recognized in linear time.\n\\end{Corollary}\n\n\\begin{Corollary}\\label{cor:p2-probeblock}\n$2$-probe block graphs are distance-hereditary.\n\\end{Corollary}\n\\proof\\, A slightly stronger statement holds. Each block of a $2$-probe block graph is clearly a $2$-probe complete graph. By Theorem~\\ref{thm:2-probecomplete}, each block of a $2$-probe block graph is therefore a cograph.\\qed\n\n\\section{Partitioned $2$-probe block graphs}\\label{sec:p2-probeblock}\nLet $G=(V,E)$ be a graph with two given independent set $\\N_1, \\N_2\\subseteq V$. Suppose there exists a set\n$E'\\subseteq \\binom{\\N_1}{2}\\cup \\binom{\\N_2}{2}$ such that the graph $G=(V,E\\cup E')$ is a block graph, that is, $G$ is a\npartitioned $2$-probe block graph with respect to the given independent sets $\\N_1, \\N_2$.\nThen, clearly, the two non-adjacent vertices $x,y$ of every induced diamond in $G$ must belong to one of $\\N_1,\\N_2$ and $\\{x,y\\}$ must belong to $E'$. Similarly, any two non-adjacent vertices $x,y$ of every induced $4$-cycle in $G$ must belong to one of $\\N_1,\\N_2$ and $\\{x,y\\}$ must belong to $E'$.\n\nIn what follows, given a graph $G=(V,E)$ together with two given independent sets $\\N_1, \\N_2$,\nthe \\emph{enhanced graph} $G^*=(V,E^*)$ is obtained from $G$ by adding all edges between two vertices both in $\\N_1$ or both in $\\N_2$ that are two vertices of an induced diamond or of an induced $C_4$ in $G$.\n\nPartitioned probe block graphs can be characterized as follows.\n\n\\begin{Theorem}\\label{thm:p2-probeblock}\nLet $G=(V,E)$ be a graph with two independent sets $\\N_1,\\N_2$ and $\\Probe=V\\setminus\\big(\\N_1\\cup\\N_2\\big)$.\nThen the following statements are equivalent:\n\\begin{itemize}\n \\item[\\em (i)]  $G=(\\Probe,\\N_1,\\N_2, E)$ is a partitioned $2$-probe block graph;\n \\item[\\em (ii)] Every block $B$ of $G$ is a $(K_B,X_B,Y_B,Z_B)$-graph such that $\\N_1\\cap B=X_B\\cup Z_B$ and $\\N_2\\cap B=Y_B\\cup Z_B$;\n \\item[\\em (iii)] The enhanced graph $G^*$ of $G$ is a block graph.\n\\end{itemize}\n\\end{Theorem}\n\\proof\n\n\\smallskip\\noindent\n(i) $\\Rightarrow$ (ii): Since every block $B$ of $G$ is contained in a block of any block graph embedding of $G$, $B$ is a partitioned $2$-probe complete graph. Hence (ii) follows by Corollary~\\ref{cor:p2-probecomplete}.\n\n\\smallskip\\noindent\n(ii) $\\Rightarrow$ (iii): Let $G=(V,E)$ satisfy (ii). Then, for every two vertices $x, y$ of $G$, we have the following fact:\n\\begin{align*}\n&\\text{Both $x$ and $y$ are in $\\N_1$ or in $\\N_2$ and belong to an induced diamond or to}\\\\\n&\\text{an induced $C_4$ if and only if $x,y$ are non-adjacent vertices in a block of $G$.}\n\\end{align*}\nTo see this, note first that one direction is obvious: every diamond and every $C_4$ is contained in a block of $G$. Conversely, let $x, y$ be two non-adjacent vertices in a block $B$ of $G$. As $G$ satisfies (ii), $B=(K_B, X_B, Y_B, Z_B)$ and hence $x,y\\in X_B\\cup Z_B\\subseteq \\N_1$ or  $x,y\\in Y_B\\cup Z_B\\subseteq \\N_2$. Moreover, as $B$ is 2-connected, it is easy to see that $x, y$ are contained in a diamond or a $C_4$ in $B$. \n\nThus, by definition of $G^*=(V,E^*)$, $xy\\in E^*\\setminus E$ if and only if $x,y$ are non-adjacent vertices of a block in $G$. Therefore, each block of $G^*$ is a clique, that is, $G^*$ is a block graph.\n\n\\smallskip\\noindent\n(iii) $\\Rightarrow$ (i): This implication is obvious.\n\\qed\n\n\\medskip\nSince the blocks of a graph can be computed in linear time, and $(K,X,Y,Z)$-graphs can be recognized in linear time, Theorem~\\ref{thm:p2-probeblock} (ii) implies:\n\\begin{Corollary}\\label{cor:recog-p2-probeblock}\nPartitioned $2$-probe block graphs can be recognized in linear time.\n\\end{Corollary}\n\n\n\\section{Unpartitioned $2$-probe block graphs}\\label{sec:2-probeblock}\nIn this section, we characterize $2$-probe block graphs in terms of their block structure and gluing conditions. The characterization reminds the one of $1$-probe block graphs (Theorem~\\ref{thm:probeblock}), but it is considerably more involved. It turns out that the blocks are $2$-probe complete graphs and can be described by six forbidden induced subgraphs depicted in Figure~\\ref{fig:B1-6}, and the gluing conditions can be expressed in terms of the other sixteen forbidden induced subgraphs depicted in Figure~\\ref{fig:G1-16}.\n\n\\begin{Theorem}\\label{thm:2-probeblock}\nThe following statements are equivalent for any graph $G$:\n\\begin{itemize}\n\\item[\\em (i)] $G$ is a $2$-probe block graph;\n\\item[\\em (ii)] $G$ is a $\\{B_1, \\ldots, B_6, G_1,\\ldots, G_{16}\\}$-free distance-hereditary graph;\n\\item[\\em (iii)] $G$ is a $\\{G_1, \\ldots, G_{16}\\}$-free\n                 graph in which every block is a $2$-probe complete graph.\n\\end{itemize}\n\\end{Theorem}\n\n\\begin{figure}[htb\n\\begin{center}\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (34) at (2,1.5) {};\n\\node[vP] (5) at (1,3) {};\n\\node[vP] (6) at (0,2) {};\n\n\\draw (1) -- (2) -- (34) -- (5) -- (6) -- (1); \\draw (1) -- (5); \\draw (2) -- (6);\n\n\\node[] (B1) at (1,0) [label=below:$B_{1}$] {}; \n\\end{tikzpicture}\n\\quad\\,\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (2,1) {};\n\\node[vP] (4) at (2,2) {};\n\\node[vP] (5) at (1,3) {};\n\\node[vP] (6) at (0,2) {};\n\n\\draw (1) -- (2) -- (3) -- (4) -- (5) -- (6) -- (1) -- (4); \\draw (3) -- (6);\n\n\\node[] (B2) at (1,0) [label=below:$B_{2}$] {}; \n\\end{tikzpicture}\n\\quad\\,\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (2,1) {};\n\\node[vP] (4) at (2,2) {};\n\\node[vP] (5) at (1,3) {};\n\\node[vP] (6) at (0,2) {};\n\n\\draw (1) -- (2) -- (3) -- (4) -- (5) -- (6) -- (1) -- (4); \\draw (3) -- (6);\n\\draw (1) -- (5) -- (3);\n\\draw (4) -- (2) -- (6);\n\n\\node[] (B3) at (1,0) [label=below:$B_{3}$] {}; \n\\end{tikzpicture}\n\\quad\\,\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (2,1) {};\n\\node[vP] (4) at (2,2) {};\n\\node[vP] (5) at (1,3) {};\n\\node[vP] (6) at (0,2) {};\n\n\\draw (1) -- (2) -- (3) -- (4) -- (5) -- (6) -- (1) -- (3) -- (6) -- (4) -- (1);\n\n\\node[] (B4) at (1,0) [label=below:$B_{4}$] {}; \n\\end{tikzpicture}\n\\quad\\,\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (2,1) {};\n\\node[vP] (4) at (2,2) {};\n\\node[vP] (5) at (1,3) {};\n\\node[vP] (6) at (0,2) {};\n\n\\draw (1) -- (2) -- (3) -- (4) -- (5) -- (6) -- (1) -- (3) -- (6) -- (4);\n\\draw (3) -- (5); \\draw (2) -- (6);\n\n\\node[] (B5) at (1,0) [label=below:$B_{5}$] {}; \n\\end{tikzpicture}\n\\quad\\,\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (2,1) {};\n\\node[vP] (4) at (2,2) {};\n\\node[vP] (5) at (1,3) {};\n\\node[vP] (6) at (0,2) {};\n\n\\draw (1) -- (2) -- (3) -- (4) -- (5) -- (6) -- (1) -- (3) -- (6) -- (4) -- (1);\n\\draw (2) -- (4); \\draw (2) -- (6);\n\n\\node[] (B6) at (1,0) [label=below:$B_{6}$] {}; \n\\end{tikzpicture}\n\\end{center}\n\\caption{Block structure: $2$-connected forbidden induced subgraphs for unpartitioned $2$-probe block graphs.}\n\\label{fig:B1-6}\n\\end{figure}\n\n\n\\proof\n\n\\smallskip\\noindent\n(i) $\\Rightarrow$ (ii): By Corollary~\\ref{cor:p2-probeblock}, $G$ is distance-hereditary. By inspection, none of $B_1$, \\ldots, $B_6$, $G_1$, \\ldots, $G_{16}$ is a $2$-probe block graph.\n\n\\smallskip\\noindent\n(ii) $\\Rightarrow$ (iii): Let $G$ satisfy (ii), and let $B$ be a block of $G$. Then $B$ is a $2$-connected distance-hereditary graph without induced $B_1$, \\ldots, $B_6$. We claim that $B$ is $\\{P_4, 2K_2, K_3+K_1\\}$-free.\n\n\\begin{figure}[H\n\\begin{center}\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5a) at (2,0) {};\n\\node[vP] (5b) at (4,0) {};\n\\node[vP] (6) at (3,2) {};\n\\node[vP] (7) at (4,1) {};\n\n\\draw (2) -- (4) -- (3) -- (2) -- (1) -- (3);\n\\draw (5a) -- (4) -- (6) -- (7) -- (5b) -- (5a) -- (7) -- (4) -- (5b);\n\n\\node[] (G1) at (2,0) [label=below:$G_1$] {}; \n\\end{tikzpicture}\n\\hfill\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5a) at (2,0) {};\n\\node[vP] (5b) at (4,0) {};\n\\node[vP] (6) at (3,2) {};\n\\node[vP] (7) at (4,1) {};\n\n\\draw (2) -- (4) -- (3) -- (1) -- (2);\n\\draw (5a) -- (4) -- (6) -- (7) -- (5b) -- (5a) -- (7) -- (4) -- (5b);\n\n\\node[] (G2) at (2,0) [label=below:$G_2$] {}; \n\\end{tikzpicture}\n\\hfill\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5a) at (2,0) {};\n\\node[vP] (5b) at (4,0) {};\n\\node[vP] (6) at (3,2) {};\n\\node[vP] (7) at (4,1) {};\n\n\\draw (2) -- (4) -- (3) -- (2) -- (1) -- (3);\n\\draw (5a) -- (4) -- (6) -- (7) -- (5b) -- (5a) -- (6); \\draw (7) -- (4) -- (5b);\n\n\\node[] (G3) at (2,0) [label=below:$G_3$] {}; \n\\end{tikzpicture}\n\\hfill\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5a) at (2,0) {};\n\\node[vP] (5b) at (4,0) {};\n\\node[vP] (6) at (3,2) {};\n\\node[vP] (7) at (4,1) {};\n\n\\draw (2) -- (4) -- (3) -- (1) -- (2);\n\\draw (5a) -- (4) -- (6) -- (7) -- (5b) -- (5a) -- (6); \\draw (7) -- (4) -- (5b);\n\n\\node[] (G4) at (2,0) [label=below:$G_4$] {}; \n\\end{tikzpicture}\n\\end{center}\n\\begin{center}\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1a) at (0,0) {};\n\\node[vP] (1b) at (0,2) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,0) {};\n\\node[vP] (6) at (3,2) {};\n\\node[vP] (7) at (4,1) {};\n\n\\draw (1a) -- (2) -- (3) -- (4) -- (5) -- (7) -- (6) -- (4) -- (2);\n\\draw (1a) -- (3); \\draw (4) -- (7); \\draw (1a) -- (1b) -- (2); \\draw (1b) -- (3);\n\n\\node[] (G5) at (2,0) [label=below:$G_5$] {}; \n\\end{tikzpicture}\n\\hfill\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1a) at (0,0) {};\n\\node[vP] (1b) at (0,2) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,0) {};\n\\node[vP] (6) at (3,2) {};\n\\node[vP] (7) at (4,1) {};\n\n\\draw (1a) -- (2) -- (3) -- (4) -- (5) -- (7) -- (6) -- (4) -- (2);\n\\draw (1a) -- (3);\n\\draw (1a) -- (1b) -- (2); \\draw (1b) -- (3);\n\n\\node[] (G6) at (2,0) [label=below:$G_6$] {}; \n\\end{tikzpicture}\n\\hfill\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1a) at (0,0) {};\n\\node[vP] (1b) at (0,2) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,0) {};\n\\node[vP] (6) at (3,2) {};\n\\node[vP] (7) at (4,1) {};\n\\node[vP] (8) at (4,2) {};\n\n\\draw (1a) -- (2) -- (3) -- (4) -- (5) -- (7) -- (6) -- (4) -- (2);\n\\draw (1a) -- (3); \\draw (4) -- (7); \\draw (1a) -- (1b) -- (2); \\draw (1b) -- (3); \n\\draw (5) -- (6) -- (8) -- (7);\n\n\\node[] (G7) at (2,0) [label=below:$G_7$] {}; \n\\end{tikzpicture}\n\\hfill\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1a) at (0,0) {};\n\\node[vP] (1b) at (0,2) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,1) {};\n\\node[vP] (6) at (4,0) {};\n\\node[vP] (7) at (4,2) {};\n\\node[vP] (8) at (5,1) {};\n\n\\draw (1a) -- (2) -- (3) -- (4) -- (5) --(6) -- (8) -- (7) -- (5); \\draw (6) -- (7);\n\\draw (1a) -- (3); \\draw (2) -- (4); \\draw (1a) -- (1b) -- (2); \\draw (1b) -- (3);\n\n\\node[] (G8) at (2.5,0) [label=below:$G_8$] {}; \n\\end{tikzpicture}\n\\end{center}\n\\begin{center}\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1a) at (0,0) {};\n\\node[vP] (1b) at (0,2) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,1) {};\n\\node[vP] (6) at (4,0) {};\n\\node[vP] (7) at (4,2) {};\n\\node[vP] (8) at (5,1) {};\n\n\\draw (1a) -- (2) -- (3) -- (4) -- (5) --(6) -- (8) -- (7) -- (5);\n\\draw (1a) -- (3); \\draw (2) -- (4); \\draw (1a) -- (1b) -- (2); \\draw (1b) -- (3);\n\n\\node[] (G9) at (2.5,0) [label=below:$G_9$] {}; \n\\end{tikzpicture}\n\\hfill\\quad\\,\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,0) {};\n\\node[vP] (6) at (3,2) {};\n\\node[vP] (7) at (4,1) {};\n\\node[vP] (8) at (5,0) {};\n\\node[vP] (9) at (5,2) {};\n\\node[vP] (10) at (6,1) {};\n\n\\draw (2) -- (4) -- (3) -- (1) -- (2) -- (3);\n\\draw (4) -- (5) -- (7) -- (6) -- (4) -- (7);\n\\draw (8) -- (10) -- (9) -- (7) -- (8) -- (9);\n\n\\node[] (G12) at (3,0) [label=below:$G_{10}$] {}; \n\\end{tikzpicture}\n\\hfill\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,0) {};\n\\node[vP] (6) at (3,2) {};\n\\node[vP] (7) at (4,1) {};\n\\node[vP] (8) at (5,0) {};\n\\node[vP] (9) at (5,2) {};\n\\node[vP] (10) at (6,1) {};\n\n\\draw (2) -- (4) -- (3) -- (1) -- (2);\n\\draw (4) -- (5) -- (7) -- (6) -- (4) -- (7);\n\\draw (8) -- (10) -- (9) -- (7) -- (8) -- (9);\n\n\\node[] (G13) at (3,0) [label=below:$G_{11}$] {}; \n\\end{tikzpicture}\n\\end{center}\n\\begin{center}\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,0) {};\n\\node[vP] (6) at (3,2) {};\n\\node[vP] (7) at (4,1) {};\n\\node[vP] (8) at (5,0) {};\n\\node[vP] (9) at (5,2) {};\n\\node[vP] (10) at (6,1) {};\n\n\\draw (2) -- (4) -- (3) -- (1) -- (2);\n\\draw (4) -- (5) -- (7) -- (6) -- (4) -- (7);\n\\draw (8) -- (10) -- (9) -- (7) -- (8);\n\n\\node[] (G14) at (3,0) [label=below:$G_{12}$] {}; \n\\end{tikzpicture}\n\\hfill\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,2) {};\n\\node[vP] (6) at (2,3) {};\n\\node[vP] (7) at (3,4) {};\n\\node[vP] (8) at (4,3) {};\n\\node[vP] (9) at (4,1) {};\n\\node[vP] (10) at (5,0) {};\n\\node[vP] (11) at (5,2) {};\n\\node[vP] (12) at (6,1) {};\n\n\\draw (2) -- (4) -- (3) -- (1) -- (2) -- (3);\n\\draw (5) -- (4) -- (9) -- (5);\n\\draw (6) -- (7) -- (8) -- (5) -- (6) -- (8);\n\\draw (11) -- (12) -- (10) -- (9) -- (11) -- (10);\n\n\\node[] (G15) at (3,0) [label=below:$G_{13}$] {}; \n\\end{tikzpicture}\n\\hfill\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,2) {};\n\\node[vP] (6) at (2,3) {};\n\\node[vP] (7) at (3,4) {};\n\\node[vP] (8) at (4,3) {};\n\\node[vP] (9) at (4,1) {};\n\\node[vP] (10) at (5,0) {};\n\\node[vP] (11) at (5,2) {};\n\\node[vP] (12) at (6,1) {};\n\n\\draw (2) -- (4) -- (3) -- (1) -- (2);\n\\draw (5) -- (4) -- (9) -- (5);\n\\draw (6) -- (7) -- (8) -- (5) -- (6) -- (8);\n\\draw (11) -- (12) -- (10) -- (9) -- (11) -- (10);\n\n\\node[] (G16) at (3,0) [label=below:$G_{14}$] {}; \n\\end{tikzpicture}\n\\end{center}\n\\vspace*{-2em}\n\\begin{center}\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,2) {};\n\\node[vP] (6) at (2,3) {};\n\\node[vP] (7) at (3,4) {};\n\\node[vP] (8) at (4,3) {};\n\\node[vP] (9) at (4,1) {};\n\\node[vP] (10) at (5,0) {};\n\\node[vP] (11) at (5,2) {};\n\\node[vP] (12) at (6,1) {};\n\n\\draw (2) -- (4) -- (3) -- (1) -- (2);\n\\draw (5) -- (4) -- (9) -- (5);\n\\draw (6) -- (7) -- (8) -- (5) -- (6);\n\\draw (11) -- (12) -- (10) -- (9) -- (11) -- (10);\n\n\\node[] (G17) at (3,0) [label=below:$G_{15}$] {}; \n\\end{tikzpicture}\n\\qquad\\qquad\n\\begin{tikzpicture}[scale=.55]\n\\tikzstyle{vP}=[circle,inner sep=1.5pt,fill=black];\n\\node[vP] (1) at (0,1) {};\n\\node[vP] (2) at (1,0) {};\n\\node[vP] (3) at (1,2) {};\n\\node[vP] (4) at (2,1) {};\n\\node[vP] (5) at (3,2) {};\n\\node[vP] (6) at (2,3) {};\n\\node[vP] (7) at (3,4) {};\n\\node[vP] (8) at (4,3) {};\n\\node[vP] (9) at (4,1) {};\n\\node[vP] (10) at (5,0) {};\n\\node[vP] (11) at (5,2) {};\n\\node[vP] (12) at (6,1) {};\n\n\\draw (2) -- (4) -- (3) -- (1) -- (2);\n\\draw (5) -- (4) -- (9) -- (5);\n\\draw (6) -- (7) -- (8) -- (5) -- (6);\n\\draw (11) -- (12) -- (10) -- (9) -- (11);\n\n\\node[] (G18) at (3,0) [label=below:$G_{16}$] {}; \n\\end{tikzpicture}\n\\end{center}\n\\caption{Gluing conditions: Forbidden induced subgraphs for unpartitioned $2$-probe block graphs.}\n\\label{fig:G1-16}\n\\end{figure}\n\n\nAssume first that $B$ contains an induced subgraph $P_4$, say $abcd$. Since $B$ is $2$-connected distance-hereditary, there is a vertex $x$ adjacent to $a$ and $c$ (hence non-adjacent to $d$), and another vertex $y$ adjacent to $b$ and $d$ (hence non-adjacent to $a$). Let $H$ be the subgraph of $B$ induced by $a,b,c,d,x$ and $y$. As $H$ is not a domino, one of the edges $xy, xb, yc$ must exist. Then, an easy case analysis shows that $H$ is isomorphic to $B_2$ or to $B_4$, or $H-a$ or $H-d$ is a house or a gem. Thus, $B$ is a cograph. Since $B$ is $2$-connected, $B=H_1\\star H_2$. Assume next that $B$ contains an induced subgraph $F\\in\\{2K_2,K_3+K_1\\}$. Then $F$ is contained in $H_1$, say, and $|V(H_2)|=1$ (otherwise there would be a $B_1$ or $B_5$ in case $F=2K_2$, or a $B_1$ or $B_6$ in case $F=K_3+K_1$). Therefore $H_1$ is connected and hence $H_1=H_{11}\\star H_{12}$ with $F$ being contained in $H_{11}$, say. But now there is a $B_5$ or a $B_6$ induced by $F$, $H_2$ and a vertex in $H_{12}$.\n\nThus, $B$ is $\\{P_4, 2K_2, K_3+K_1\\}$-free, as claimed. Note that $B_1=(K_2+K_1)\\star 2K_1$ and $B_3=2K_1\\star 2K_1\\star 2K_1$.\nHence, by Theorem~\\ref{thm:2-probecomplete}, $B$ is a $2$-probe complete graph.\n\n\\smallskip\\noindent\n(iii) $\\Rightarrow$ (i): We prove a slightly stronger claim that every graph $H$ satisfying (iii) admits two (possibly empty) independent sets $\\N_1$ and $\\N_2$ such that\n\\begin{equation}\\label{eq1}\n\\text{$H=(\\Probe,\\N_1, \\N_2,E)$ is a partitioned $2$-probe block graph,}\n\\end{equation}\n\\begin{equation}\\label{eq2}\n\\begin{split}\n&\\text{$\\forall\\, i=1,2$, $\\forall\\, v\\in\\N_i$, there is another vertex $v'\\in\\N_i$ such that $v$}\\\\\n&\\text{and $v'$ are degree-$2$ vertices of an induced $C_4$ or diamond in $H$,}\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{eq3}\n\\begin{split}\n&\\text{every vertex $v\\in\\N_1\\cap \\N_2$ is the degree-$2$ vertex of some}\\\\\n&\\text{induced $F_1$ (see Figure~\\ref{fig:F1234}) in $H$,}\n\\end{split}\n\\end{equation}\nand,\n\\begin{equation}\\label{eq4}\n\\begin{split}\n&\\text{$\\forall\\, v\\in\\Probe$, $\\forall\\, x\\in N(v)\\cap\\N_1, \\forall\\, y\\in N(v)\\cap \\N_2$, it holds that:}\\\\\n&\\text{$x\\in\\N_2$ or $y\\in\\N_1$ or $xy\\in E(H)$.}\n\\end{split}\n\\end{equation}\n\nWe will prove this claim by induction. Let $G$ satisfy (iii).\nIf $G$ is itself a block, then by assumption, $G$ is a $2$-probe complete graph, and by Theorem~\\ref{thm:2-probecomplete}, $G$ is a $(K,X,Y,Z)$-graph. If $G$ is a clique, set $\\N_1=\\N_2=\\emptyset$. If $X\\not=\\emptyset$ and $Y\\not=\\emptyset$, set $\\N_1=X\\cup Z$, $\\N_2=Y\\cup Z$. Finally, if $X=\\emptyset$ or $Y=\\emptyset$, set $\\N_1=X\\cup Y\\cup Z$, $\\N_2=\\emptyset$. It is clear, by the $2$-connectedness of $G$, the properties (\\ref{eq1}), (\\ref{eq2}), (\\ref{eq3}) and (\\ref{eq4}) hold in this case.\n\nSo, consider an end-block $B$ of $G$ and let $v$ be the cut-vertex of $G$ in $B$. Let $H=G-(V(B)\\setminus\\{v\\})$. By induction, $H$ admits independent sets $\\N_1', \\N_2'$ satisfying (\\ref{eq1}), (\\ref{eq2}), (\\ref{eq3}) and (\\ref{eq4}).\n\nIf $B$ is a clique, then clearly $\\N_1:= \\N_1'$ and $\\N_2:=\\N_2'$ are independent sets of $G$ satisfying (\\ref{eq1}), (\\ref{eq2}), (\\ref{eq3}) and (\\ref{eq4}), and we are done.\n\nSo, we may assume that $B$ is not a clique. By assumption, $B$ is a $2$-probe complete graph. Write $B=(K,X,Y,Z)$, where $K$ is the set of all universal vertices of $B$ and $Z$ is the set of all isolated vertices of $B-K$. Then, as $B$ is $2$-connected and not complete, \n\\begin{itemize}\n\\item $Z\\not=\\emptyset$ and $|K|\\ge 2$, or else \n\\item $|X|\\ge 2$ and $|Y|\\ge 2$. \n\\end{itemize}\nMoreover, by definition of $Z$ and $K$, \n\\begin{itemize}\n\\item $X=\\emptyset$ if and only if $Y=\\emptyset$, and \n\\item if $Z=\\emptyset$, then $|X|\\ge 2, |Y|\\ge 2$.\n\\end{itemize}\n\n\\smallskip\\noindent\n\\textsc{Case 1.}\\, $v\\in K$.\\, \nIn this case, $v$ belongs to an induced diamond $D$ in $B$ with $\\deg_D(v)=3$. This can be seen as follows: In case $|X|\\ge 2$ and $|Y|\\ge 2$, $v$, two vertices in $X$ and a vertex in $Y$ together induced such a diamond $D$. In other case, $Z\\not=\\emptyset$ and $|K|\\ge 2$. Let $w$ be a vertex in $K\\setminus\\{v\\}$. If $|Z|\\ge 2$, then $v, w$ and two vertices in $Z$ induce such a diamond $D$. If $|Z|=1$, then $X\\not=\\emptyset$ (as $B$ is not complete) and $v, w$, the vertex in $Z$ and a vertex in $X$ induce such a diamond $D$.  \n\nAssume first that $v\\in\\N_1'\\cap\\N_2'$. By (\\ref{eq3}), $v$ is the degree-$2$ vertex of some induced $F_1$ in $H$. Then, as the cut-vertex $v$ is the only common vertex of this $F_1$ and $D$, this $F_1$ and $D$ together induce a $G_5$, a contradiction. Thus, $v\\not\\in\\N_1'\\cap\\N_2'$.\n\nAssume next that $v\\in\\N_1'\\cup\\N_2'$, say $v\\in\\N_1'$ but $v\\not\\in \\N_2'$. Then, by\n(\\ref{eq2}), $v$ belongs to a $4$-vertex induced subgraph $H'$ in $H$ that is a $C_4$ or a diamond with $\\deg_{H'}(v)=2$. Hence $Z\\not=\\emptyset$  (otherwise $H'$, two vertices in $X$ and two vertices in $Y$ together would induce a $G_3$ or $G_4$), and $X=\\emptyset$ and $Y=\\emptyset$ (otherwise $H'$, a vertex in $Z$, a vertex in $K\\setminus\\{v\\}$, a vertex in $X$ and a vertex in $Y$ together would induce a $G_1$ or $G_2$). Then, as $v\\not\\in \\N_2'$, $\\N_1:= \\N_1'$ and $\\N_2:= \\N_2'\\cup Z$ are independent sets in $G$ satisfying (\\ref{eq1}), (\\ref{eq2}), (\\ref{eq3}) and (\\ref{eq4}), and we are done.\n\nThus, we may assume that $v\\not\\in\\N_1'\\cup\\N_2'$. In this case, define $\\N_1$ and $\\N_2$ as follows. If $X\\not=\\emptyset$ and $Y\\not=\\emptyset$, then $\\N_1:=\\N_1'\\cup X\\cup Z$ and $\\N_2:=\\N_2'\\cup Y\\cup Z$. Otherwise (recall that $X=\\emptyset$ if and only if $Y=\\emptyset$), $\\N_1:=\\N_1'\\cup Z$ and $\\N_2:=\\N_2'$. Clearly, $\\N_1$ and $\\N_2$ are independent sets of $G$ satisfying (\\ref{eq1}), (\\ref{eq2}), (\\ref{eq3}) and (\\ref{eq4}).\n\nCase 1 is settled.\n\n\\medskip\\noindent\n\\textsc{Case 2}\\, $v\\not\\in K$.\\, \nIn this case, $v$ belongs to a $4$-vertex induced subgraph $F$ in $B$ such that $F$ is a diamond or a $C_4$ and $\\deg_F(v)=2$. Moreover, if $Z=\\emptyset$, then $F=C_4$. This can be seen as follows: Suppose first $v\\in X\\cup Y$, say $v\\in X$. If $Z=\\emptyset$, then $|X|\\ge 2, |Y|\\ge 2$, and $v$, another vertex in $X$ and two vertices in $Y$ together induce such an $F=C_4$. If $Z\\not=\\emptyset$, then $|K|\\ge 2$, and $v$, two vertices in $K$ and a vertex in $Z$ induce such a diamond $F$. Suppose next $v\\in Z$. If $|Z|\\ge 2$, then $v$, another vertex in $Z$ and two vertices in $K$ together induce such a diamond $F$. If $Z=\\{v\\}$, then $X\\not=\\emptyset$ (as $B$ is not complete), and $v$, two vertices in $K$ and a vertex in $X$ together induce such a diamond $F$.  \n\nWe first show that\n\\begin{equation*}\n\\text{$v$ has no neighbor in $\\N_1'\\cap N_2'$.}\n\\end{equation*}\nFor otherwise, let $x\\in N(v)\\cap\\N_1'\\cap N_2'$. By (\\ref{eq3}), $x$ belongs to an induced subgraph $H'=F_1$ in $H$ with $\\deg_{H'}(x)=2$. Let $B'$ be the block of $H$ containing $H'$. Now, if $v\\not\\in B'$, then $F$ and $H'$ together induce a $G_8$ or $G_9$.\nSo, let $v\\in B'$. Then, by (\\ref{eq1}), $v$ is a universal vertex in $B'$ (as $v\\not\\in\\N_1'\\cup\\N_2'$) and hence $v$ is not one of the two degree-$3$ vertices $y_1, y_2$ of $H'$ (which belong to $\\N_1'\\cup\\N_2'$).\nThus, $v\\not\\in H'$ or $v$ is one of the degree-$4$ vertices of $H'$. Then $F, x, y_1, y_2$ and a degree-$4$ vertex in $H'$ different from $v$ together induce a $G_{1}$ or $G_{2}$. In any case, we have a contradiction, hence $v$ cannot have a neighbor in $\\N_1'\\cap N_2'$, as claimed.\n\n\\medskip\nNext we show that\n\\begin{equation*}\n\\text{$N(v)\\cap \\N_1'=\\emptyset$ or $N(v)\\cap \\N_2'=\\emptyset$.}\n\\end{equation*}\nFor otherwise let $x_1\\in N(v)\\cap \\N_1'$ and $x_2\\in N(v)\\cap \\N_2'$. Since $v$ has no neighbor in $\\N_1'\\cap \\N_2'$, $x_1\\not\\in\\N_2', x_2\\not\\in\\N_1'$ (in particular, $x_1\\not=x_2$). Hence, by (\\ref{eq4}), $x_1x_2\\in E(H)$. By (\\ref{eq2}), $x_i$ belongs to a $4$-vertex induced subgraph $H_i$ in $H$ that is a $C_4$ or a diamond with $\\deg_{H_i}(x_i)=2$, $i=1,2$, and the vertex $x_i'$ in $H_i$ non-adjacent to $x_i$ also belongs to $\\N_i'$. Let $B_i$ be the blocks of $H$ containing $H_i$. Now, if $v\\not\\in B_1\\cup B_2$, then $B_1\\cap B_2=\\emptyset$ and $F$, $H_1, H_2$ together induce a $G_{13}, G_{14}, G_{15}$ or $G_{16}$.\nIf $v\\in B_1\\cap B_2$, then in particular $B_1= B_2$, and by (\\ref{eq1}), $v$ is a universal vertex in $B_1=B_2$, hence (as $v$ has no neighbor in $\\N_1'\\cap \\N_2'$), $x_1'\\not\\in\\N_2'$, $x_2'\\not\\in\\N_1'$. Therefore, by (\\ref{eq1}), $x_1, x_2, x_1', x_2'$ must induce a $C_4$ in $B_1=B_2$. But then $F$ and $x_1, x_2, x_1', x_2'$ together induce a $G_3$ or $G_4$.\nSo, let $v\\in B_1$ and $v\\not\\in B_2$, say. Then $B_1\\cap B_2=\\{x_2\\}$.\nAs before, by (\\ref{eq1}), $v$ is a universal vertex in $B_1$, hence $x_1'\\not\\in\\N_2'$. Therefore $x_1'$ must be adjacent to $x_2$. But then $F$, $x_1, x_1'$, and $H_2$ together induce a $G_{10}, G_{11}$ or $G_{12}$. In any case, we have a contradiction, hence $v$ cannot have neighbors in both $\\N_1'$ and $\\N_2'$, as claimed.\n\nWe now distinguish two subcases.\n\n\\medskip\\noindent\n\\textsc{Subcase 2.1.}\\, $N(v)\\cap\\N_1'=N(v)\\cap \\N_2'=\\emptyset$.\n\nAssume first that $v\\in\\N_1'\\cap \\N_2'$ and $v\\not\\in Z$. Then $v\\in X\\cup Y$ and thus both $X$ and $Y$ are nonempty; let $v\\in X$, say. By (\\ref{eq3}), $v$ is the degree-$2$ vertex of some induced $H'=F_1$ in $H$. Now, if $Z=\\emptyset$ then $F$ is a $C_4$ and therefore $F$ and $H'$ together induce a $G_6$. If $Z\\not=\\emptyset$ then a vertex in $Z$, two vertices in $K$, a vertex in $Y$ and $H'$ together induce a $G_7$. In any case we have a contradiction.\n\nThus, $v\\in Z$ or $v\\not\\in\\N_1'$ or $v\\not\\in \\N_2'$. Now, define $\\N_1$ and $\\N_2$ as follows.\n\nSuppose $v\\in Z$. If $X\\not=\\emptyset$ and $Y\\not=\\emptyset$, $\\N_1:=\\N_1'\\cup X\\cup Z$ and $\\N_2:=\\N_2'\\cup Y\\cup Z$. Otherwise, $\\N_1:=\\N_1'\\cup Z$ and $\\N_2:=\\N_2'$.\n\nSuppose $v\\not\\in Z$ and say $v\\not\\in\\N_1'$. If $v\\in X$, $\\N_1:=\\N_1'\\cup Y\\cup Z$ and $\\N_2:=\\N_2'\\cup X\\cup Z$. Otherwise, $\\N_1:=\\N_1'\\cup X\\cup Z$ and $\\N_2:=\\N_2'\\cup Y\\cup Z$.\n\nThen, as $N(v)\\cap\\N_1'=N(v)\\cap \\N_2'=\\emptyset$, $\\N_1$ and $\\N_2$ are independent sets of $G$, and clearly, satisfy (\\ref{eq1}), (\\ref{eq2}), (\\ref{eq3}) and (\\ref{eq4}).\n\n\\medskip\\noindent\n\\textsc{Subcase 2.2.}\\, $N(v)\\cap\\N_1'=\\emptyset$ and $N(v)\\cap \\N_2'\\not=\\emptyset$, say. Then $v\\not\\in\\N_2'$.\n\nAssume first that $v\\in Z$. Then $X=Y=\\emptyset$. Otherwise, consider a vertex $x\\in N(v)\\cap \\N_2'$. By (\\ref{eq2}), $x$ belongs to a $4$-vertex induced subgraph $H'$ in $H$ that is a $C_4$ or a diamond with $\\deg_{H'}(x)=2$. Let $B'$ be the block of $H$ containing $H'$.\nIf $v\\not\\in B'$, then $v$, two vertices in $K$, a vertex in $X$, a vertex in $Y$ and $H'$ together induce a $G_8$ or $G_9$.\nIf $v\\in B'$, then by (\\ref{eq1}), $v$ is a universal vertex in $B'$, or $v\\in\\N_1'$ and $H'$ is a $C_4$. Recall that the vertex $x'$ in $H'$ nonadjacent to $x$ also belongs to $\\N_2'$. Let $y$ be a vertex in $H'-\\{x,x',v\\}$. Note that $v, x,y,x'$ induce a diamond in $B'$ (if $v$ is universal in $B'$) or a $C_4$ (if $v,y\\in\\N_1'$). Now, two vertices in $K$, a vertex in $X$, a vertex in $Y$ and $v,x,y,x'$ together induce a $G_5$ or $G_6$.\n\nThus, if $v\\in Z$ then $X=Y=\\emptyset$, and, as $N(v)\\cap\\N_1'=\\emptyset$, $\\N_1:=\\N_1'\\cup Z$ and $\\N_2:=\\N_2'$ are independent sets of $G$ satisfying (\\ref{eq1}), (\\ref{eq2}), (\\ref{eq3}) and (\\ref{eq4}).\n\nSo, we may assume that $v\\in X\\cup Y$. Then, if $v\\in X$, set $\\N_1:=\\N_1'\\cup X\\cup Z$ and $\\N_2:=\\N_2'\\cup Y\\cup Z$. If $v\\in Y$, set $\\N_1:=\\N_1'\\cup Y\\cup Z$ and $\\N_2:=\\N_2'\\cup X\\cup Z$. It is clear that, as $N(v)\\cap\\N_1'=\\emptyset$, in any case, $\\N_1$ and $\\N_2$ are independent sets of $G$ satisfying (\\ref{eq1}), (\\ref{eq2}), (\\ref{eq3}) and (\\ref{eq4}).\n\n\\medskip\nCase 2 is settled. The proof of (iii) $\\Rightarrow$ (i) is complete, hence Theorem~\\ref{thm:2-probeblock}.\\qed\n\n\\section{A linear time recognition of unpartitioned $2$-probe block graphs}\\label{sec:reog}\nBased on Theorem~\\ref{thm:2-probeblock} and its proof, we describe briefly in this section how to recognize in linear time whether a given graph is a $2$-probe block graph, and if so, to output a partition into probes and non-probes.\n\n\\begin{enumerate}\n\\item\\label{step1}\nCompute the blocks and the cut-vertices of $G$.\n\\item\\label{step2}\nFor each non-complete block $B$ of $G$, let $K_B$ be the set of its universal vertices and $Z_B$ be the set of all isolated vertices of $B-K_B$.\\\\\nIf $B-(K_B\\cup Z_B)$ is not a complete bipartite graph, then output ``NO'' and STOP, meaning that $G$ is not a $2$-probe block graph.\\\\\nOtherwise, let $(X_B,Y_B)$ be the bipartition of the complete bipartite graph; possibly empty.\\\\\nLet $\\mathcal{B}$ be the set of all blocks $B=(K_B, X_B,Y_B,Z_B)$ and their cut-vertices of $G$.\n\\item\\label{step3}\nFollow the proof of Theorem~\\ref{thm:2-probeblock}, part (iii) $\\Rightarrow$ (i), to compute the candidates $\\N_1$ and $\\N_2$.\n\\item\\label{step4}\nCheck if $\\N_1$ and $\\N_2$ are independent sets of $G$. If not, output ``NO'' and STOP. Otherwise check if $G=(\\Probe,\\N_1,\\N_2,E)$ is a partitioned $2$-probe block graph. If this is the case, output ``YES'', meaning $G$ is a $2$-probe block graph. If not, output ``NO''.\n\\end{enumerate}\n\nGiven Theorem~\\ref{thm:2-probeblock} and its proof, the correctness is clear. It is also clear that each step, except step~\\ref{step3}, can be implemented with linear time complexity.\nStep~\\ref{step3} can be described more precisely by the following procedure \\texttt{FindNonprobes}.\n\n\\begin{table}[hbt]\n  \\begin{center}\n    Procedure \\texttt{FindNonprobes}$(\\mathcal{B}; \\N_1,\\N_2)$\\\\[.7ex]\n    \\fbox{\\,\n       \\begin{minipage}{\\textwidth}\n        \\begin{tabbing}\n          \\OUTPUT\\=\\kill\n          \\INPUT \\>  set $\\mathcal{B}$ of blocks.\\\\\n          \\OUTPUT\\>  sets of vertices $\\N_1, \\N_2$.\\\\[1ex]\n          (N.)\\hspace*{-1.5ex}\\=\\ei\\=\\ei\\=\\ei\\=\\ei\\=\\ei\\=\\ei\\=\\ei\\=\\ei\\=\\ei\\=\\ei\\kill\n          1.\\>\\> \\IF $\\mathcal{B}=\\{B\\}$ \\THEN\\\\\n          2.\\>\\>\\> if $B$ is a clique, set $\\N_1:=\\emptyset; \\N_2:= \\emptyset$.\\\\\n          3.\\>\\>\\> if $X_B\\not=\\emptyset$, set $\\N_1:=X_B\\cup Z_B; \\N_2:= Y_B\\cup Z_B$.\\\\\n          4.\\>\\>\\> if $X_B=Y_B=\\emptyset$, set $N_1:= Z_B, \\N_2:=\\emptyset$.\\\\\n         \n          5.\\>\\> \\ELSE\\\\\n          6.\\>\\>\\> let $B\\in\\mathcal{B}$ be an end-block with cut-vertex $v$.\\\\\n          7.\\>\\>\\> set $\\mathcal{B}':= \\mathcal{B}\\setminus\\{B\\}$ and call \\texttt{FindNonprobes}$(\\mathcal{B}'; \\N_1',\\N_2')$.\\\\\n          8.\\>\\>\\> \\IF $B$ is a clique \\THEN\\\\\n          9.\\>\\>\\>\\> $\\N_1:=\\N_1'; \\N_2:= \\N_2'$\\\\\n        \n         10.\\>\\>\\> \\ELSE\\\\\n         11.\\>\\>\\>\\>  follow the proof of Theorem~\\ref{thm:2-probeblock} (iii) $\\Rightarrow$ (i).\\\\\n         12.\\>\\>\\>\\>  if $v\\in K_B$, compute $\\N_1$ and $\\N_2$ according to Case 1.\\\\\n         13.\\>\\>\\>\\>  if $v\\not\\in K_B$, compute $\\N_1$ and $\\N_2$ according to Case 2.\\\\\n         14.\\>\\>\\> \\ENDIF\\\\\n         15.\\>\\> \\ENDIF\n        \\end{tabbing}\n      \\end{minipage}\n    \\quad}\n  \\end{center}\n  \\label{alg:FindNonprobes}\n\\end{table}\n\nLet $t(\\mathcal{B})$ denote the time needed by \\texttt{FindNonprobes}$(\\mathcal{B}; \\N_1,\\N_2)$. Note that each block $B$ has $|E(B)|\\ge |V(B)|$, unless $B\\in\\{K_1,K_2\\}$. Thus, we have $t(\\{B\\})=O(|E(B)|)+O(1)$ and $t(\\mathcal{B})=t(\\mathcal{B}\\setminus\\{B\\})+O(|E(B)|)+O(1)$. Therefore, $t(\\mathcal{B})=\\sum_{B\\in\\mathcal{B}}( O(|E(B)|)+O(1))= O(|E(G)|+|V(G)|)$.\nTo sum up, we have:\n\\begin{Theorem}\\label{thm:reg-2-probeblock}\nUnpartitioned $2$-probe block graphs can be recognized in linear time.\n\\end{Theorem}\n\nWe note that our algorithm is optimal in the following senses: If $G$ is a $2$-probe block graph, the partition into $(\\Probe,\\N_1,\\N_2)$ is minimal, {\\em i.e.}, the corresponding block graph embedding has minimal number of new edges. Moreover, if $G$ is a $1$-probe block graph, then the algorithm will output $\\N_2=\\emptyset$.\n\n\\section{Conclusion}\\label{sec:conclusion}\nWith Theorems \\ref{thm:2-probecomplete}, \\ref{thm:p2-probeblock}, and~\\ref{thm:2-probeblock} we have given good characterizations of $2$-probe complete graphs and $2$-probe block graphs. This might be a\nfirst step towards the solution of the challenging problems of characterizing and recognizing $k$-probe complete graphs and $k$-probe block graphs for any $k\\ge 3$.\n\nThese problems seem be very difficult even for certain restricted graph classes.\nFor instance, it is not clear which cographs are $k$-complete graphs, given $k$.\n\n\\medskip\\noindent\n\\textbf{Problem 1.}\n{\\em Let $k\\ge 3$. Characterize cographs that are $k$-probe complete graphs, respectively, $k$-probe block graphs.\n}\n\n\\medskip\\noindent\nNote that any graph $G=(V,E)$ is a $k$-probe complete graph for some $k$; for instance, $k=\\binom{|V|}{2} - |E|$.\nIn~\\cite{ChaHunKloPen}, it is proved that determining the smallest integer $k$ such that\n$G$ is a $k$-probe complete graph is NP-hard. Indeed, it is equivalent to determine the smallest number of\ncliques in $\\overline{G}$ that cover the edges of $\\overline{G}$, which is a well-known NP-hard problem~\\cite{Orlin77}.\n\nWe remark that, in connection to Problem~1, it is still unknown how, given a cograph $G$, to compute the smallest integer $k$ such that $G$ is a $k$-probe complete graph. Indeed, Ton Kloks posed the following problem in 2007.\n\n\\medskip\\noindent\n\\textbf{Problem 2 (Kloks~\\cite{Kloks07,Kloks-cstheory})}\n{\\em\nGiven a cograph $G$, determine the smallest integer $k$ such that $G$ is a $k$-probe complete graph.\n}\n\n\\medskip\\noindent\nSince cographs are self-complementary, Kloks' problem is equivalent to\ndetermine the minimum number of cliques that cover the edges of a given cograph $G$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe ARCH and GARCH models have become important tools in time series analysis. The ARCH model has been introduced by Engle in \\cite{En} and then it has been generalized by Bollerslev to the GARCH model in \\cite{Bol1}. Since, a large collection of variants and extensions of these models has been  produced by many authors.  See for example \\cite{Bol2}  for a glossary of models derived from ARCH and GARCH. \n\nIn this work, we also focus on a generalized ARCH model, namely the model (\\ref{arch}).  Our contribution proposes to include in the expression of the squared volatility $\\sigma _{t} ^{2} $ a factor $L_{t-1}$, which we will call {\\it liquidity.} The motivation to consider such a model comes from mathematical finance, where the factor $L_{t}$, which constitutes a proxi for the trading volume at day $t$, has been included in order to capture the fluctuations of the intra-day price in financial markets. A more detailed explanation can be found in  \\cite{BTT} or \\cite{TT}. In the work \\cite{BTT} we considered the particular case when $L_{t}$ is the squared increment of the fractional Brownian motion (fBm in the sequel), i.e. $L_{t}= (B ^{H}_{t+1} -B^{H}_{t}) ^{2} $, where $B ^{H}$ is a fBm with Hurst parameter $ H\\in (0,1)$. \n\nIn this work, our purpose is twofold. Firstly, we enlarge the ARCH with fBm liquidity in \\cite{BTT} by considering, as a proxi for the liquidity, a general positive (strictly) stationary process $(L_{t}) _{t\\in \\mathbb{Z}}$.  This includes, besides the above mentioned case of the squared increment of the fBm,  many other examples. \n\n The second purpose is to provide a method to estimate the parameters of the model.  As mentioned in \\cite{BTT},  in the case when $L$ is a process without independent increments, the usual approaches for the parameter estimation in ARCH models (such as least squares method and maximum likelihood method) do not work, in the sense that the estimators obtained by these classical methods are biased and not consistent. Here we adopt a different technique, based on the AR(1) characterization of the ARCH process, which has also been used in \\cite{vouti}. The AR(1) characterization leads to Yule-Walker type equations for the parameters of the model. These equations are of quadratic form and  then we are able to find explicit formulas for the estimators. We prove that the estimators are consistent by using extended version of the law of large numbers and by assuming enough regularity for the correlation structure of the liquidity process. We also provide a numerical analysis of the estimators.\n\nThe rest of the paper is organised as follows. In Section \\ref{sec:model} we introduce our model and prove the existence and uniqueness of the stationary solution. We also provide necessary and sufficient conditions for the existence of the autocovariance function. We derive the AR(1) characterization and  Yule-Walker type equations for the parameters of the model.  Section \\ref{sec:estimation} is devoted to the estimation of the model parameters.  We construct estimators in a closed form and we prove their consistency via extended versions of the law of large numbers and a control of the behaviour of the covariance of the liquidity process.  Several examples are discussed in details. In particular, we study squared increments of the fBm, squared increments of the compensated Poisson process, and the squared increments of the Rosenblatt process.\nWe end the paper with a numerical analysis of our estimators.\n\n\n\n\\section{The model}\n\\label{sec:model}\nThe generalized ARCH model is defined for every $t\\in\\mathbb{Z}$ as \n\n\\begin{equation}\n\\label{arch}\nX_t = \\sigma_t\\epsilon_t, \\qquad \\sigma_t^2 = \\alpha_0+\\alpha_1X_{t-1}^2 + l_1L_{t-1},\n\\end{equation}\nwhere $\\alpha_0\\geq 0$, $\\alpha_1,l_1>0$, and $(\\epsilon_t)_{t\\in\\mathbb{Z}}$ is an i.i.d. process with $\\mathbb{E}(\\epsilon_0) = 0$ and $\\mathbb{E}(\\epsilon_0^2)=1$. Moreover, $(L_t)_{t\\in\\mathbb{Z}}$ is a strictly stationary positive process with $\\mathbb{E}(L_0) = 1$ and independent of $(\\epsilon_t)_{t\\in\\mathbb{Z}}$. We first give sufficient conditions to ensure the existence of a stationary solution.\n\\noindent\nNote that we have a recursion\n\n\\begin{equation}\n\\label{recursiveformula}\n\\sigma_t^ 2 = \\alpha_0 + \\alpha_1 \\epsilon_{t-1}^ 2 \\sigma_{t-1}^ 2 + l_1 L_{t-1}.\n\\end{equation}\nLet us denote\n\n\\begin{equation*}\nA_t = \\alpha_1\\epsilon_t^ 2\\quad\\text{and} \\quad B_t = \\alpha_0 + l_1L_t \\quad\\text{for every }t\\in\\mathbb{Z}.\n\\end{equation*}\nUsing \\eqref{recursiveformula} $k+1$ times we get\n\n\\begin{equation}\n\\label{iteration}\n\\begin{split}\n\\sigma_{t+1}^ 2 &= A_t \\sigma_t^ 2 + B_t\\\\\n&= A_tA_{t-1} \\sigma_{t-1}^ 2 + A_tB_{t-1} + B_t\\\\\n&=\\hdots\\\\\n&= \\left(\\prod_{i=0}^k A_{t-i}\\right) \\sigma_{t-k}^ 2 + \\sum_{i=0}^ k\\left(\\prod_{j=0}^ {i-1} A_{t-j}\\right) B_{t-i},\n\\end{split}\n\\end{equation}\nwith the convention $\\prod_0^ {-1} = 1$.\n\nThe following lemma ensures that we are able to continue the recursion infinitely many times.\n\\begin{lemma}\n\\label{lemma:stationarity}\nSuppose $\\alpha_1<1$ and $\\sup_{t\\in\\mathbb{Z}}\\mathbb{E}(\\sigma_t^ 2) \\leq M_1<\\infty$. Then, as $k\\to \\infty$, we have\n\n\\begin{equation*}\n \\left(\\prod_{i=0}^k A_{t-i}\\right) \\sigma_{t-k}^ 2 \\to 0\n\\end{equation*}\nin $L^ 1$. Furthermore, if $\\alpha_1 <  \\frac{1}{\\sqrt{\\mathbb{E}(\\epsilon_0^ 4)}}$ and $\\sup_{t\\in\\mathbb{Z}}\\mathbb{E}(\\sigma_t^4)\\leq M_2 < \\infty$, then the convergence holds also almost surely.\n\n\\begin{proof}\nBy independence of $\\epsilon$, we have\n\\begin{equation*}\n\\begin{split}\n\\mathbb{E}\\left| \\left(\\prod_{i=0}^k A_{t-i}\\right) \\sigma_{t-k}^ 2\\right| = \\alpha_1^ {k+1} \\mathbb{E}(\\sigma_{t-k}^ 2)\\leq \\alpha_1^ {k+1} M_1\\to 0\n\\end{split}\n\\end{equation*}\nproving the first part of the claim. \nFor the second part, Chebysev's inequality implies\n\n\\begin{footnotesize}\n\\begin{equation*}\n\\begin{split}\n\\mathbb{P}\\left(\\left|\\left(\\prod_{i=0}^k A_{t-i}\\right) \\sigma_{t-k}^ 2 - \\alpha_1^ {k+1}\\mathbb{E}(\\sigma_{t-k}^ 2)\\right|>\\varepsilon\\right) &\\leq \\frac{\\mathrm{Var}\\left(\\left(\\prod_{i=0}^k A_{t-i}\\right) \\sigma_{t-k}^ 2\\right)}{\\varepsilon^ 2}\\\\\n&=\\frac{\\alpha_1^ {2k+2} \\mathbb{E}\\left(\\left(\\prod_{i=0}^ k \\epsilon^ 4_{t-i}\\right) \\sigma_{t-k}^ 4\\right) - \\alpha_1^ {2k+2} \\mathbb{E}(\\sigma_{t-k}^2)^2}{\\varepsilon^ 2} \\\\\n&\\leq \\frac{\\left(\\alpha_1^ 2 \\mathbb{E}(\\epsilon^ 4_0)\\right)^ {k+1} M_2 - \\alpha_1^ {2k+2} M_1^2}{\\varepsilon^ 2},\n\\end{split}\n\\end{equation*}\n\\end{footnotesize}\nwhich is summable by assumptions. Borel-Cantelli then implies \n\n\\begin{equation*}\n\\left(\\prod_{i=0}^k A_{t-i}\\right) \\sigma_{t-k}^ 2 - \\alpha_1^ {k+1}\\mathbb{E}(\\sigma_{t-k}^ 2)\\to 0\n\\end{equation*}\nalmost surely proving the claim.\n\\end{proof}\n\\end{lemma}\n\n\\subsection{Existence of a stationary solution}\nThe following theorem gives the existence of a stationary solution under relatively weak assumptions (we only assume the existence of the second moment of $L$ and the usual condition $\\alpha_{1} <1 $ (see e.g. \\cite{FZ})). \n\n\\begin{theorem}\n\\label{theo:stationarity}\nAssume that $\\mathbb{E}(L_0^ 2) < \\infty$ and  $\\alpha_1< 1$. Then \\eqref{arch} has the following strictly stationary solution \n\n\\begin{equation}\n\\label{uniquesolution}\n\\sigma_{t+1}^ 2 = \\sum_{i=0}^ \\infty\\left(\\prod_{j=0}^ {i-1} A_{t-j}\\right) B_{t-i}.\n\\end{equation}\n\n\\begin{proof}\nWe begin by showing that \\eqref{uniquesolution} is well-defined. That is, we prove that \n\n\\begin{equation*}\n\\lim_{k\\to\\infty} \\sum_{i=0}^ k\\left(\\prod_{j=0}^ {i-1} A_{t-j}\\right) B_{t-i}\n\\end{equation*}\ndefines an almost surely finite random variable. \nFirst we observe that the summands above are non-negative and hence, the pathwise limits exist in $[0,\\infty]$. Write\n\n\\begin{equation}\n\\label{twoseries}\n\\sum_{i=0}^ \\infty\\left(\\prod_{j=0}^ {i-1} A_{t-j}\\right) B_{t-i} = \\alpha_0 \\sum_{i=0}^ \\infty \\left(\\prod_{j=0}^ {i-1} A_{t-j}\\right) + l_1 \\sum_{i=0}^ \\infty \\left(\\prod_{j=0}^ {i-1} A_{t-j}\\right) L_{t-i}\n\\end{equation}\nand denote\n\n\\begin{equation*}\na_n =\\left (\\prod_{j=0}^ {n-1} A_{t-j}\\right) L_{t-n}, \\quad b_n =\\left (\\prod_{j=0}^ {n-1} A_{t-j}\\right).\n\\end{equation*}\nBy the root test it suffices to prove that \n\\begin{equation}\n\\label{eq:root-a}\n\\limsup_{n\\to\\infty} a_n^ {\\frac{1}{n}} < 1\n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:root-b}\n\\limsup_{n\\to\\infty} b_n^ {\\frac{1}{n}} < 1.\n\\end{equation}\nHere\n\n\\begin{equation*}\na_n^{\\frac{1}{n}} = e^{\\frac{1}{n}\\log a_n} = L_{t-n}^ {\\frac{1}{n}} e^{\\frac{1}{n} \\sum_{j=0}^{n-1} \\log A_{t-j}},\n\\end{equation*}\nwhere\n\n\\begin{equation*}\ne^{\\frac{1}{n} \\sum_{j=0}^{n-1} \\log A_{t-j}} \\overset{a.s}{\\longrightarrow}  e^{\\mathbb{E}\\log A_0} = \\alpha_1 e^{\\mathbb{E} \\log \\epsilon_0^ 2}\n\\end{equation*}\nby the law of large numbers and continuous mapping theorem. By Jensen's inequality we obtain that\n\n\\begin{equation*}\n \\alpha_1 e^{\\mathbb{E} \\log \\epsilon_0^ 2} \\leq \\alpha_1 e^{\\log \\mathbb{E}(\\epsilon_0^ 2)} = \\alpha_1 < 1.\n\\end{equation*}\nThat is \n\n\\begin{equation*}\n\\lim_{n\\to\\infty} e^{\\frac{1}{n} \\sum_{j=0}^{n-1} \\log A_{t-j}} < 1\n\\end{equation*}\nalmost surely. This proves \\eqref{eq:root-b} which implies that the first series in \\eqref{twoseries} is almost surely convergent. To obtain \\eqref{eq:root-a}, it remains to show $\\limsup_{n\\to\\infty} L_{t-n}^ {\\frac{1}{n}} \\leq 1$ almost surely. We have\n\n\\begin{equation}\n\\label{indicators}\nL_{t-n}^ {\\frac{1}{n}} = \\mathbbm{1}_{L_{t-n} < 1}L_{t-n}^ {\\frac{1}{n}} + \\mathbbm{1}_{L_{t-n} \\geq 1} L_{t-n}^ {\\frac{1}{n}} \\leq 1 + \\mathbbm{1}_{L_{t-n} \\geq 1} L_{t-n}^ {\\frac{1}{n}} - \\mathbbm{1}_{L_{t-n} \\geq 1} \n\\end{equation}\nwhere we have used\n$$\n\\mathbbm{1}_{L_{t-n} < 1}L_{t-n}^ {\\frac{1}{n}} \\leq \\mathbbm{1}_{L_{t-n} < 1} = 1 - \\mathbbm{1}_{L_{t-n} \\geq 1}.\n$$\nNow\n\n\\begin{equation*}\n\\begin{split}\n\\mathbb{P}\\left(\\left|\\mathbbm{1}_{L_{t-n} \\geq 1} L_{t-n}^ {\\frac{1}{n}} - \\mathbbm{1}_{L_{t-n} \\geq 1}\\right| \\geq \\varepsilon\\right) &\\leq \\frac{\\mathbb{E}\\left|\\mathbbm{1}_{L_{t-n} \\geq 1} L_{t-n}^ {\\frac{1}{n}} - \\mathbbm{1}_{L_{t-n} \\geq 1}\\right|^ 2}{\\varepsilon^ 2}\\\\\n&= \\frac{\\mathbb{E}\\left(\\mathbbm{1}_{L_{t-n} \\geq 1}\\left(L_{t-n}^ {\\frac{1}{n}} -1\\right)^ 2\\right)}{\\varepsilon^ 2}.\n\\end{split}\n\\end{equation*}\nConsider now the function $f_x(a) \\coloneqq x^ a$ for  $x\\geq 1$ and $a\\geq 0$. Since $f_x'(a) = x^a\\log x$ we obtain by the mean value theorem that\n\n\\begin{equation*}\n\\left|f_x(a) - f_x(0)\\right| \\leq \\max_{0\\leq b \\leq a} \\left|f_x'(b)\\right|a = ax^a\\log x.\n\\end{equation*}\nHence\n\n\\begin{equation*}\n\\mathbbm{1}_{L_{t-n} \\geq 1} \\left(L_{t-n}^ {\\frac{1}{n}} -1\\right)^ 2 \\leq\\mathbbm{1}_{L_{t-n} \\geq 1}  \\frac{1}{n^ 2}L_{t-n}^ {\\frac{2}{n}} \\left(\\log L_{t-n}\\right)^ 2.\n\\end{equation*}\nOn the other hand, for $n\\geq 2$ and $L_{t-n} \\geq 1$ it holds that\n\n\\begin{equation*}\n\\frac{ L_{t-n}^ {\\frac{2}{n}} \\left(\\log L_{t-n}\\right)^ 2}{L_{t-n}^ 2} \\leq  \\frac{ \\left(\\log L_{t-n}\\right)^ 2}{L_{t-n}}< 1,\n\\end{equation*}\nsince for $x\\geq 1$, the function $g(x) \\coloneqq \\left(\\log x\\right)^ 2x^ {-1}$ has the maximum $g(e^2) = 4e^ {-2}$. \nConsequently,\n\n\\begin{equation*}\n \\frac{\\mathbb{E}\\left(\\mathbbm{1}_{L_{t-n} \\geq 1}\\left(L_{t-n}^ {\\frac{1}{n}} -1\\right)^ 2\\right)}{\\varepsilon^ 2} < \\frac{\\mathbb{E}\\left(\\mathbbm{1}_{L_{t-n} \\geq 1} L_{t-n}^ 2\\right)}{\\varepsilon^ 2 n^ 2} \\leq \\frac{\\mathbb{E}\\left(L_{t-n}^ 2\\right)}{\\varepsilon^ 2 n^ 2}.\n\\end{equation*}\nHence Borel-Cantelli implies\n\n\\begin{equation*}\n\\mathbbm{1}_{L_{t-n} \\geq 1} L_{t-n}^{\\frac{1}{n}} - \\mathbbm{1}_{L_{t-n} \\geq 1} \\overset{\\text{a.s.}}{\\to} 0\n\\end{equation*}\nwhich by \\eqref{indicators} shows \\eqref{eq:root-a}. Let us next show that \\eqref{uniquesolution} satisfies \\eqref{recursiveformula}.\n\n\\begin{equation*}\n\\begin{split}\nA_t \\sigma_t^ 2 + B_t &= \\sum_{i=0}^\\infty \\left(\\prod_{j=0}^i A_{t-j}\\right) B_{t-i-1} + B_t\\\\\n&= \\sum_{i=1}^\\infty \\left(\\prod_{j=0}^{i-1} A_{t-j}\\right) B_{t-i} + B_t\\\\\n&=  \\sum_{i=0}^ \\infty\\left(\\prod_{j=0}^ {i-1} A_{t-j}\\right) B_{t-i} = \\sigma_{t+1}^ 2.\n\\end{split}\n\\end{equation*}\nIt remains to prove that \\eqref{uniquesolution} is stationary. However, since $(A_t, B_t)$ is stationary, we have\n\n\\begin{equation*}\n\\sum_{i=0}^ k\\left(\\prod_{j=0}^ {i-1} A_{t-j}\\right) B_{t-i} \\overset{\\text{law}}{=} \\sum_{i=0}^ k\\left(\\prod_{j=0}^ {i-1} A_{-j}\\right) B_{-i}\n\\end{equation*}\nfor every $t$ and $k$. Since the limits of the both sides exist as $k\\to\\infty$ we have\n\n\\begin{equation*}\n\\sigma_{t+1}^ 2 = \\sum_{i=0}^ \\infty\\left(\\prod_{j=0}^ {i-1} A_{t-j}\\right) B_{t-i} \\overset{\\text{law}}{=} \\sum_{i=0}^ \\infty\\left(\\prod_{j=0}^ {i-1} A_{-j}\\right) B_{-i} = \\sigma_1^ 2.\n\\end{equation*}\nTreating multidimensional distributions similarly concludes the proof.\n\\end{proof}\n\\end{theorem}\n\nWe show below that the stationary solution is unique in some class of processes.\n\n\n\\begin{kor}\n\\label{kor:unique-stat}\nSuppose $\\alpha_1 < 1$ and $\\mathbb{E}(L_0^2)<\\infty$. Then \\eqref{arch} has a unique solution given by \\eqref{uniquesolution} in the class of processes satisfying $\\sup_{t\\in\\mathbb{Z}}\\mathbb{E} (\\sigma_t^2) < \\infty$. \n\\end{kor}\n\\begin{proof}\nBy Theorem \\ref{theo:stationarity} \\eqref{uniquesolution} provides a stationary solution. Hence it remains to prove the uniqueness. \nBy \\eqref{iteration} we have for every $t\\in\\mathbb{Z}$ and $k\\in\\{0,1,\\hdots\\}$ that\n\\begin{equation*}\n\\sigma_{t+1}^2 =  \\left(\\prod_{i=0}^k A_{t-i}\\right) \\sigma_{t-k}^ 2 + \\sum_{i=0}^ k\\left(\\prod_{j=0}^ {i-1} A_{t-j}\\right) B_{t-i}.\n\\end{equation*}\nSuppose now that there exists two solutions $\\sigma_{t}^2$ and $\\tilde{\\sigma}_t^2$ satisfying $\\sup_{t\\in\\mathbb{Z}}\\mathbb{E}(\\sigma_t^2) < \\infty$ and $\\sup_{t\\in\\mathbb{Z}}\\mathbb{E}(\\tilde{\\sigma}_t^2) < \\infty$. Then\n$$\n|\\sigma_{t+1}^2 - \\tilde{\\sigma}_{t+1}^2| \\leq  \\left(\\prod_{i=0}^k A_{t-i}\\right) \\sigma_{t-k}^ 2 + \\left(\\prod_{i=0}^k A_{t-i}\\right) \\tilde{\\sigma}_{t-k}^ 2.\n$$\nAs both terms on the right-side converges in $L^1$ to zero by Lemma \\ref{lemma:stationarity}, we observe that \n$$\n\\mathbb{E}|\\sigma_{t+1}^2 - \\tilde{\\sigma}_{t+1}^2| = 0\n$$\nfor all $t\\in\\mathbb{Z}$ which implies the result.\n\\end{proof}\n\n\\begin{rem}\nWe assumed that the liquidity $(L_{t})_{t\\in \\mathbb{Z}}$ is a strictly stationary sequence. Nevertheless, the results in this section can be obtained by assuming that $(L_{t})_{t\\in \\mathbb{Z}}$ is weakly stationary (i.e., we have the shift-invariance in time of the first and second moments of the process). That is, by assuming weak stationarity of the noise, we obtain weak stationarity of the volatility $(\\sigma _{t} ^{2} )_{t\\in \\mathbb{Z}}$ in Theorem \\ref{theo:stationarity}. We prefer to keep the assumption of strict stationarity because it is needed later to simplify the third and fourth order assumptions of Lemma \\ref{lemma:consistency2} and also because our main examples of liquidities are strictly stationary processes (see Section \\ref{examples})\n\n\\end{rem}\n\nIn the sequel, we consider the stationary solution $(\\sigma^2_t)_{t\\in\\mathbb{Z}}$ given by Theorem \\ref{theo:stationarity}. Therefore, we will always implicitly assume that \n\n$$\\mathbb{E}(L_0^2) <\\infty \\mbox{ and  }\\alpha_1<1.$$ \n\nIn order to study covariance function of the solution \\eqref{uniquesolution}, we need that the moments $\\mathbb{E} (\\sigma_t^4)$ exists. Necessary and sufficient conditions for this are given in the following lemma. \n\n\\begin{lemma}\n\\label{thefourthmoment}\nSuppose $\\mathbb{E}(\\epsilon_0^4) < \\infty$. Then $\\mathbb{E}(\\sigma^4_0) < \\infty$ if and only if $\\alpha_1 < \\frac{1}{\\sqrt{\\mathbb{E}(\\epsilon_0^4)}}$.\n\\end{lemma}\n\\begin{proof}\nDenote $\\mathbb{E}(\\epsilon_0^4) = C_\\epsilon$ and $\\mathbb{E}(L_0^2) = C_L$. By the definition \\eqref{uniquesolution} of the strictly stationary solution\n\\begin{equation*}\n\\mathbb{E}(\\sigma_{t+1}^4) = \\mathbb{E}\\left(\\sum_{i=0}^{\\infty} \\left(\\prod_{j=0}^{i-1} A_{t-j}\\right)B_{t-i}\\right)^2,\n\\end{equation*}\nand since all the terms above are positive, both sides are simultaneously finite or infinite. Note also that, as the terms all positive, we may apply Tonelli's theorem to change the order of summation and integration obtaining\n\n\\begin{equation}\n\\label{diagonals}\n\\begin{split}\n\\mathbb{E}(\\sigma_{t+1}^4) &= \\sum_{i=0}^{\\infty} \\mathbb{E}\\left(\\left(\\prod_{j=0}^{i-1} A_{t-j}\\right)^2B_{t-i}^2\\right)\\\\\n&\\ + \\sum_{\\mathclap{\\begin{subarray}{c}\ni,k=0\\\\ \ni \\neq k\n\\end{subarray}}}^{\\infty} \\mathbb{E} \\left(\\left(\\prod_{j=0}^{i-1} A_{t-j}\\right)B_{t-i}\\left(\\prod_{j=0}^{k-1} A_{t-j}\\right)B_{t-k}\\right).\n\\end{split}\n\\end{equation}\nLet us begin with the first term above. By independence, we obtain \n\n\\begin{equation}\n\\label{firstanal}\n\\begin{split}\n\\sum_{i=0}^{\\infty} \\mathbb{E}\\left(\\left(\\prod_{j=0}^{i-1} A_{t-j}^2\\right)B_{t-i}^2\\right) &= \\sum_{i=0}^{\\infty} \\left(\\prod_{j=0}^{i-1} \\alpha_1^2 C_\\epsilon\\right)\\mathbb{E}(B_{t-i}^2)\\\\\n &= \\mathbb{E}(B_0^2)\\sum_{i=0}^{\\infty} (\\alpha_1^{2} C_\\epsilon)^i.\n\\end{split}\n\\end{equation}\nConsequently, $\\mathbb{E}(\\sigma_0^4) < \\infty$ implies $\\alpha_1 < \\frac{1}{\\sqrt{C_\\epsilon}}$, since it is the radius of convergence of the series above. For the converse, consider the latter term in \\eqref{diagonals}. By Cauchy-Schwarz inequality we obtain\n\n\\begin{equation*}\n\\begin{split}\n&\\ \\sum_{\\mathclap{\\begin{subarray}{c}\ni,k=0\\\\ \ni \\neq k\n\\end{subarray}}}^{\\infty} \\mathbb{E} \\left(\\left(\\prod_{j=0}^{i-1} A_{t-j}\\right)B_{t-i}\\left(\\prod_{j=0}^{k-1} A_{t-j}\\right)B_{t-k}\\right)\\\\\n&\\leq \\sum_{\\mathclap{\\begin{subarray}{c}\ni,k=0\\\\ \ni \\neq k\n\\end{subarray}}}^{\\infty}\\sqrt{\\mathbb{E}\\left(\\left(\\prod_{j=0}^{i-1} A_{t-j}^2\\right)B_{t-i}^2\\right)}  \\sqrt{\\mathbb{E}\\left(\\left(\\prod_{j=0}^{k-1} A_{t-j}^2\\right)B_{t-k}^2\\right)}\\\\\n&=\\mathbb{E}(B_0^2)\\sum_{\\mathclap{\\begin{subarray}{c}\ni,k=0\\\\ \ni \\neq k\n\\end{subarray}}}^{\\infty}  (\\alpha_1^2 C_\\epsilon)^{\\frac{i}{2}} (\\alpha_1^2 C_\\epsilon)^{\\frac{k}{2}},\n\\end{split}\n\\end{equation*}\nwhere\n\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\mathclap{\\begin{subarray}{c}\ni,k=0\\\\ \ni \\neq k\n\\end{subarray}}}^{\\infty}  (\\alpha_1^2 C_\\epsilon)^{\\frac{i}{2}} (\\alpha_1^2 C_\\epsilon)^{\\frac{k}{2}} &<  \\sum_{i=0}^\\infty (\\alpha_1 C_\\epsilon^{\\frac{1}{2}})^i\\sum_{k=0}^\\infty (\\alpha_1  C_\\epsilon^{\\frac{1}{2}})^k.\n\\end{split}\n\\end{equation*}\nTogether with \\eqref{firstanal} this shows that if $\\alpha_1 < \\frac{1}{\\sqrt{C_\\epsilon}}$, all the series are convergent and thus $\\mathbb{E}(\\sigma_0^4) < \\infty$.\n\\end{proof}\n\\begin{rem}\nAs expected, in order to have finite moments of higher order we needed to pose more restrictive assumption $\\alpha_1 < \\frac{1}{\\sqrt{\\mathbb{E}(\\epsilon_0^4)}} \\leq 1$ as $\\mathbb{E} (\\epsilon_0^2) = 1$. For example, in the case of Gaussian innovations we obtain  the well-known condition $\\alpha_1 < \\frac{1}{\\sqrt{3}}$ (see e.g. \\cite{FZ} or \\cite{Li}). An explicit expression of the fourth moment can be obtained when $L$ is the squared increment of fBm (see Lemma 4 in \\cite{BTT}).\n\\end{rem}\n\n\n\\subsection{Computation of the model parameters}\n\\label{subsec:parameters}\n\nIn this section we compute the parameters $\\alpha_{0}, \\alpha _{1}, l_{1}$ in (\\ref{arch}) by using the aucovariance functions of $X^{2}$ and $L$. To this end, we use an AR(1) characterization of the ARCH process. From this characterization, we derive, using an idea from \\cite{vouti},  a Yule -Walker equation of quadratic form for the parameters, that we can solve explicitly.  This constitutes the basis of the construction of the estimators in the next section.  From \\eqref{arch} it follows that if $(\\sigma^2_t)_{t\\in\\mathbb{Z}}$ is stationary, then so is $(X^2_t)_{t\\in\\mathbb{Z}}$. In addition \n\\begin{equation}\n\\label{X_t}\n\\begin{split}\nX_t^2 &= \\sigma_t^2\\epsilon_t^2 - \\sigma_t^2 + \\alpha_0+\\alpha_1X_{t-1}^2+l_1L_{t-1}\\\\\n&= \\alpha_0 + \\alpha_1X_{t-1}^2+ \\sigma_t^2(\\epsilon_t^2-1) + l_1L_{t-1}.\n\\end{split}\n\\end{equation}\nNow\n\n\\begin{equation*}\n\\mathbb{E}(X_t^2) = \\alpha_0+ \\alpha_1 \\mathbb{E}(X_{t-1}^2) + l_1\n\\end{equation*}\nand hence\n\n\\begin{equation}\n\\label{meanofX}\n\\mathbb{E}(X_t^2) = \\frac{\\alpha_0 + l_1}{1-\\alpha_1}.\n\\end{equation}\nLet us define an auxiliary process $(Y_t)_{t\\in\\mathbb{Z}}$ by\n\n\\begin{equation*}\nY_t = X_t^2 - \\frac{\\alpha_0+l_1}{1-\\alpha_1}.\n\\end{equation*}\nNow $Y$ is a zero-mean stationary process satisfying\n\n\\begin{equation}\n\\label{Y_t}\n\\begin{split}\nY_t &= \\alpha_1Y_{t-1} +\\alpha_0+\\sigma_t^2(\\epsilon_t^2-1) +  l_1L_{t-1} -\\frac{\\alpha_0+l_1}{1-\\alpha_1} + \\alpha_1\\frac{\\alpha_0+l_1}{1-\\alpha_1}\\\\\n&=  \\alpha_1Y_{t-1} +\\sigma_t^2(\\epsilon_t^2-1) +  l_1(L_{t-1} -1).\n\\end{split}\n\\end{equation}\nBy denoting \n\\begin{equation*}\nZ_t = \\sigma_t^2(\\epsilon_t^2-1) + l_1(L_{t-1} -1) \n\\end{equation*}\nwe may write\n\n\\begin{equation*}\nY_t = \\alpha_1Y_{t-1} + Z_t\n\\end{equation*}\ncorresponding to the AR$(1)$ characterization (\\cite{vouti}) of $Y_t$ for $0 <\\alpha_1 <1$.\\\\ \nIn what follows, we denote the autocovariance functions of $X^2$ and $L$ with $\\gamma(n)=\\mathbb{E} (X_{t} ^{2} X_{t+n} ^{2}) - ( \\frac{\\alpha_0 + l_1}{1-\\alpha_1})^2$ and $s(n)= \\mathbb{E} (L_{n} L_{t+n} ) - 1$ respectively.\n\n\n\n\\begin{lemma}\n\\label{lemma:equations}\nSuppose $\\mathbb{E}(\\epsilon_0^4) < \\infty$ and $\\alpha_1 < \\frac{1}{\\sqrt{\\mathbb{E}(\\epsilon_0^4)}}$. Then for any $n\\neq 0$ we have\n\n\\begin{equation}\n\\label{quadratic}\n\\alpha_1^2\\gamma(n) - \\alpha_1(\\gamma(n+1) + \\gamma(n-1)) + \\gamma(n) - l_1^2s(n) = 0\n\\end{equation}\nand for $n=0$ it holds that\n\n\\begin{equation}\n\\label{quadratic2}\n\\alpha_1^2 \\gamma(0) - 2\\alpha_1\\gamma(1) + \\gamma(0) - \\frac{\\mathbb{E}(X_0^4) Var(\\epsilon_0^2)}{\\mathbb{E}(\\epsilon_0^4)} - l_1^2 s(0) =0.\n\\end{equation}\n\n\\begin{proof}\nFirst we notice that\n\\begin{equation}\n\\label{fourthofX}\n\\mathbb{E}(X_0^4) = \\mathbb{E}(\\sigma_0^4\\epsilon_0^4) = \\mathbb{E}(\\sigma_0^4)\\mathbb{E}(\\epsilon_0^4) < \\infty\n\\end{equation}\nby Lemma \\ref{thefourthmoment}. Hence, the stationary processes $Y$ and $Z$ have finite second moments. Furthermore, the covariance of $Y$ coincides with the one of $X^2$. Applying Lemma 1 of \\cite{vouti} we get \n\n\\begin{equation*}\n\\alpha_1^2\\gamma(n) - \\alpha_1(\\gamma(n+1) + \\gamma(n-1)) + \\gamma(n) - r(n) = 0\n\\end{equation*}\nfor every $n\\in\\mathbb{Z}$, where $r(\\cdot)$ is the autocovariance function of $Z$. For $r(n)$ with $n\\geq 1$ we obtain\n\n\\begin{equation}\n\\label{r(n)}\n\\begin{split}\nr(n) &= \\mathbb{E}(Z_1Z_{n+1})\\\\\n&= \\mathbb{E}[(\\sigma_1^2(\\epsilon_1^2-1) + l_1(L_0-1))(\\sigma_{n+1}^2(\\epsilon_{n+1}^2-1)+l_1(L_n-1))]\\\\\n&= l_1^2\\mathbb{E}[(L_0-1)(L_n-1)] = l_1^2s(n),\n\\end{split}\n\\end{equation}\nsince the sequences $(\\epsilon_t)_{t\\in\\mathbb{Z}}$ and $(L_t)_{t\\in\\mathbb{Z}}$ are independent of each other, and $\\epsilon_t$ is independent of $\\sigma_s$ for $s\\leq t$. By the same arguments, for $n=0$ we have\n\n\\begin{equation}\n\\label{r(0)}\n\\begin{split}\nr(0) &= \\mathbb{E}\\left[\\left(\\sigma_1^2(\\epsilon_1^2-1) + l_1(L_0-1)\\right)^2\\right]\\\\ \n&= \\mathbb{E}\\left[\\sigma_1^4(\\epsilon_1^2-1)^2\\right] + l_1^2\\mathbb{E}\\left[(l_0-1)^2\\right]\\\\\n&= \\mathbb{E}(\\sigma_1^4) Var(\\epsilon_0^2) + l_1^2 s(0).\n\\end{split}\n\\end{equation}\nNow using \\eqref{fourthofX} and $\\gamma(-1) = \\gamma(1)$ completes the proof.\n\n\\end{proof}\n\\end{lemma}\n\n\n\n\\noindent\nNow, let first $n\\in\\mathbb{Z}$ with $n\\neq0$. Then\n\n\\begin{align}\n\\label{twoquadratic2}\n\\alpha_1^2 \\gamma(0) - 2\\alpha_1\\gamma(1) + \\gamma(0) - \\frac{\\mathbb{E}(X_0^4) Var(\\epsilon_0^2)}{\\mathbb{E}(\\epsilon_0^4)} - l_1^2 s(0) =0 \\nonumber\\\\\n\\alpha_1^2\\gamma(n) - \\alpha_1(\\gamma(n+1) + \\gamma(n-1)) + \\gamma(n) - l_1^2s(n) = 0.\n\\end{align}\nFrom the first equation we get\n\n\\begin{equation*}\nl_1^2 = \\frac{1}{s(0)}\\left(\\alpha_1^2 \\gamma(0) - 2\\alpha_1 \\gamma(1)+ \\gamma(0) - \\frac{\\mathbb{E}(X_0^4) Var(\\epsilon_0^2)}{\\mathbb{E}(\\epsilon_0^4)} \\right).\n\\end{equation*}\nSubstitution to \\eqref{twoquadratic2} yields\n\n\\begin{scriptsize}\n\\begin{equation*}\n\\alpha_1^2\\left(\\gamma(n) - \\frac{s(n)}{s(0)}\\gamma(0)\\right) + \\alpha_1\\left(2 \\frac{s(n)}{s(0)} \\gamma(1) - (\\gamma(n+1) + \\gamma(n-1))\\right) + \\gamma(n) + \\frac{s(n)}{s(0)} \\left(\\frac{\\mathbb{E}(X_0^4) Var(\\epsilon_0^2)}{\\mathbb{E}(\\epsilon_0^4)} - \\gamma(0) \\right) = 0\n\\end{equation*}\n\\end{scriptsize}\nLet us denote ${\\pmb{\\gamma}}_0 = [\\gamma(n+1), \\gamma(n), \\gamma(n-1), \\gamma(1), \\gamma(0), \\mathbb{E}(X_0^4)]$ and\n\n\\begin{equation}\n\\begin{split}\n\\label{quadraticterms3}\na_0({\\pmb{\\gamma}}_0) &= \\gamma(n) - \\frac{s(n)}{s(0)}\\gamma(0)\\\\\nb_0({\\pmb{\\gamma}}_0) &= 2 \\frac{s(n)}{s(0)} \\gamma(1) - (\\gamma(n+1) + \\gamma(n-1))\\\\\nc_0({\\pmb{\\gamma}}_0) &= \\gamma(n) + \\frac{s(n)}{s(0)} \\left(\\frac{\\mathbb{E}(X_0^4) Var(\\epsilon_0^2)}{\\mathbb{E}(\\epsilon_0^4)} - \\gamma(0) \\right).\n\\end{split}\n\\end{equation}\nAssuming that $a_0({\\pmb{\\gamma}}_0) \\neq 0$ we have the following solutions for the model parameters $\\alpha_1$ and $l_1$:\n\n\\begin{equation}\n\\label{alphaofgamma2}\n\\alpha_1({\\pmb{\\gamma}}_0) = \\frac{-b_0({\\pmb{\\gamma}}_0) \\pm \\sqrt{b_0({\\pmb{\\gamma}}_0)^2 - 4a_0({\\pmb{\\gamma}}_0)c_0({\\pmb{\\gamma}}_0)}}{2a_0({\\pmb{\\gamma}}_0)}\n\\end{equation}\nand\n\n\n\\begin{equation}\n\\label{lofgamma2}\nl_1({\\pmb{\\gamma}}_0) = \\sqrt{\\frac{1}{s(0)}\\left(\\alpha_1({\\pmb{\\gamma}}_0)^2 \\gamma(0) - 2\\alpha_1({\\pmb{\\gamma}}_0) \\gamma(1)+ \\gamma(0) - \\frac{\\mathbb{E}(X_0^4) Var(\\epsilon_0^2)}{\\mathbb{E}(\\epsilon_0^4)} \\right)}.\n\\end{equation}\nFinally, denoting $\\mu = \\mathbb{E}(X_0^2)$ and using \\eqref{meanofX} we may write\n\n\\begin{equation}\n\\label{alpha0ofgamma2}\n\\alpha_0({\\pmb{\\gamma}}_0, \\mu) = \\mu(1-\\alpha_1({\\pmb{\\gamma}}_0)) - l_1({\\pmb{\\gamma}}_0).\n\\end{equation}\nNow, let $n_1, n_2 \\in\\mathbb{Z}$ with $n_1\\neq n_2$ and $n_1,n_2\\neq 0$. Then\n\n\\begin{align}\n\\label{twoquadratic}\n\\alpha_1^2\\gamma(n_1) - \\alpha_1(\\gamma(n_1+1) + \\gamma(n_1-1)) + \\gamma(n_1) - l_1^2s(n_1)  &= 0\\\\\n\\alpha_1^2\\gamma(n_2) - \\alpha_1(\\gamma(n_2+1) + \\gamma(n_2-1)) + \\gamma(n_2) - l_1^2s(n_2)  &= 0. \\nonumber\n\\end{align}\nAssuming that $n_2$ is chosen in such a way that $s(n_2) \\neq 0$ we have\n\n\n\\begin{equation}\n\\label{l_1}\nl_1^2 = \\frac{\\alpha_1^2\\gamma(n_2) - \\alpha_1(\\gamma(n_2+1) + \\gamma(n_2-1)) + \\gamma(n_2)}{s(n_2)}.\n\\end{equation}\nSubstitution to \\eqref{twoquadratic} yields\n\n\n\\begin{tiny}\n\\begin{equation*}\n\\alpha_1^2\\left(\\gamma(n_1)-\\frac{s(n_1)}{s(n_2)}\\gamma(n_2)\\right) - \\alpha_1\\left(\\gamma(n_1+1) + \\gamma(n_1-1)- \\frac{s(n_1)}{s(n_2)}\\left(\\gamma(n_2+1) + \\gamma(n_2-1)\\right)\\right) + \\gamma(n_1) - \\frac{s(n_1)}{s(n_2)}\\gamma(n_2) = 0.\n\\end{equation*}\n\\end{tiny}\nLet us denote ${\\pmb{\\gamma}} = [\\gamma(n_1+1), \\gamma(n_2+1), \\gamma(n_1), \\gamma(n_2), \\gamma(n_1-1), \\gamma(n_2-1)]$ and\n\n\\begin{align}\n\\label{quadraticterms2}\n\\begin{split}\na({\\pmb{\\gamma}}) &= \\gamma(n_1) - \\frac{s(n_1)}{s(n_2)}\\gamma(n_2)\\\\\nb({\\pmb{\\gamma}}) &=  \\frac{s(n_1)}{s(n_2)}\\left(\\gamma(n_2+1) + \\gamma(n_2-1)\\right) - (\\gamma(n_1+1) + \\gamma(n_1-1)).\\\\\n\\end{split}\n\\end{align}\nAssuming $a({\\pmb{\\gamma}}) \\neq 0$ we obtain the following solutions for the model parameters $\\alpha_1$ and $l_1$:\n\n\n\\begin{equation}\n\\label{alphaofgamma}\n\\alpha_1({\\pmb{\\gamma}}) = \\frac{-b({\\pmb{\\gamma}}) \\pm \\sqrt{b({\\pmb{\\gamma}})^2 - 4a({\\pmb{\\gamma}})^2}}{2a({\\pmb{\\gamma}})},\n\\end{equation} \nand\n\n\\begin{equation}\n\\label{lofgamma}\nl_1({\\pmb{\\gamma}}) = \\sqrt{\\frac{\\alpha_1^2({\\pmb{\\gamma}})\\gamma(n_2) - \\alpha_1({\\pmb{\\gamma}})(\\gamma(n_2+1) + \\gamma(n_2-1)) + \\gamma(n_2)}{s(n_2)}}.\n\\end{equation}\nAgain, $\\alpha_0$ is given by\n\n\n\\begin{equation}\n\\label{alpha0ofgamma}\n\\alpha_0({\\pmb{\\gamma}}, \\mu) = \\mu(1-\\alpha_1({\\pmb{\\gamma}})) - l_1({\\pmb{\\gamma}}).\n\\end{equation}\n\n\n\\begin{rem}\nNote that here we assumed $s(n_2)\\neq 0$ and $a({\\pmb{\\gamma}}) \\neq 0$ which means that we choose $n_1,n_2$ in a suitable way. Notice however, that these assumptions are not a restriction. Firstly, the case where $s(n_2)=0$ for all $n_2\\neq 0$ corresponds to the more simple case where $L$ is a sequence of uncorrelated random variables. Secondly, if $s(n_2)\\neq 0$ and $a({\\pmb{\\gamma}})=0$, the second order term vanishes and \nwe get a linear equation for $\\alpha_1$. For detailed discussion on this phenomena, we refer to \\cite{vouti}.\n\\end{rem}\n\n\n\\begin{rem}\n\\label{rem:sign}\nAt first glimpse Equations \\eqref{alphaofgamma2} and \\eqref{alphaofgamma} may seem useless as one needs to choose between signs. However, it usually suffices to know additional values of the covariance of the noise (see \\cite{vouti}). In particular, it suffices that $s(n) \\to 0$ (see \\cite{vouti2}).\n\\end{rem}\n\n\n\n\\section{Parameter estimation}\n\\label{sec:estimation}\nIn this section we discuss how to estimate the model parameters consistently from the observations provided that the covariance of the liquidity $L$ is known. Based on formulas for the parameters provided in Subsection \\ref{subsec:parameters}, it suffices that the covariances of $X^2$ can be estimated consistently. \n\\subsection{Consistency of autocovariance estimators}\nThroughout this section we denote \n\\begin{equation*}\nf(t-s) = \\mathbb{E}(L_tL_s) = Cov(L_t,L_s) + 1 = s(t-s)+1.\n\\end{equation*}\n\n\\begin{lemma}\n\\label{covariance}\nLet $t,s \\in\\mathbb{Z}$. Then\n\n\\begin{equation*}\n\\mathbb{E}(\\sigma_t^ 2 L_s) = \\frac{\\alpha_0}{1-\\alpha_1} + l_1\\sum_{i=0}^ \\infty \\alpha_1^ i f(t-s-i-1)\n\\end{equation*}\n\\begin{proof}\nBy \\eqref{uniquesolution} and Fubini-Tonelli\n\n\\begin{equation*}\n\\begin{split}\n\\mathbb{E}(\\sigma_t^ 2 L_s) &= \\sum_{i=0}^\\infty\\left(\\prod_{j=0}^{i-1} \\alpha_1 \\mathbb{E}(\\epsilon_{t-1-j}^ 2)\\right)  \\mathbb{E}\\left((\\alpha_0+l_1L_{t-1-i})L_s\\right)\\\\\n&= \\alpha_0 \\sum_{i=0}^ \\infty \\alpha_1^ i + l_1\\sum_{i=0}^ \\infty \\alpha_1^ i \\mathbb{E}(L_{t-1-i} L_s)\\\\\n&=  \\alpha_0 \\sum_{i=0}^ \\infty \\alpha_1^ i + l_1\\sum_{i=0}^ \\infty \\alpha_1^i f(t-s-i-1),\n\\end{split}\n\\end{equation*}\nwhere the series converges since $\\alpha_1 <1$ and $\\mathbb{E}(L_0^2)< \\infty$.\n\\end{proof}\n\n\\end{lemma}\n\n\n\nThe following variant of the law of large number is needed for the proof of the consistency of the estimators. \n\n\n\\begin{lemma}\n\\label{lawoflarge}\nLet $(U_1, U_2, ...)$ be a sequence of random variables with a mutual expectation. In addition, assume that $\\mathrm{Var}(U_j) \\leq C$ and $\\left|\\mathrm{Cov}(U_j,U_k)\\right| \\leq g(|k-j|)$, where $g(i)\\to 0$ as $i\\to\\infty$. Then \n\n\\begin{equation*}\n\\frac{1}{n} \\sum_{k=1}^ n U_k \\to \\mathbb{E}(U_1)\n\\end{equation*}\nin probability.\n\\begin{proof}\n\nBy Chebyshev's inequality\n\\begin{equation*}\n\\begin{split}\n\\mathbb{P}\\left(\\left|\\frac{1}{n} \\sum_{k=1}^ n U_k - \\mathbb{E}(U_1)\\right| > \\varepsilon\\right)  &\\leq \\frac{\\mathrm{Var}\\left( \\sum_{k=1}^ n U_k\\right)}{\\varepsilon^2 n^2},\n\\end{split}\n\\end{equation*}\nwhere\n\n\\begin{equation*}\n\\begin{split}\n\\mathrm{Var}\\left(\\sum_{k=1}^ n U_k\\right) &= \\sum_{k,j=1}^n \\mathrm{Cov}\\left(U_k, U_j\\right)\\\\\n&= \\sum_{k=1}^n \\mathrm{Var}(U_k) + 2 \\sum_{k=1}^ n \\sum_{j=1}^{k-1} \\mathrm{Cov} (U_k,U_j)\\\\\n&\\leq nC + 2 \\sum_{k=1}^ n \\sum_{j=1}^{k-1}\\left| \\mathrm{Cov} (U_k,U_j)\\right|.\n\\end{split}\n\\end{equation*}\nFix $\\delta >0$. Then, there exists $N_\\delta\\in\\mathbb{N}$ such that $g(|k-j|) <\\delta$ whenever $|k-j| \\geq N_\\delta$. Note also that by Cauchy-Schwarz it holds that $\\left|\\mathrm{Cov}(U_k,U_j)\\right| \\leq C$. Assume that $n> N_\\delta$. Now\n\n\\begin{equation*}\n\\begin{split}\n\\sum_{k=1}^ n \\sum_{j=1}^{k-1}\\left| \\mathrm{Cov} (U_k,U_j)\\right| &\\leq \\sum_{k=1}^n \\sum_{j=1}^ {k-N_\\delta} g(|k-j|) + \\sum_{k=1}^n \\sum_{j=k-N_\\delta +1}^ {k-1}\\mathclap{C}\\\\\n&\\leq n^2\\delta + nN_\\delta C.\n\\end{split}\n\\end{equation*}\nHence\n\n\\begin{equation*}\n\\mathbb{P}\\left(\\left|\\frac{1}{n} \\sum_{k=1}^ n U_k - \\mathbb{E}(U_1)\\right| > \\varepsilon\\right) \\leq \\frac{nC + 2n^ 2\\delta + 2nN_\\delta C}{\\varepsilon^2 n^ 2} = \\frac{2\\delta}{\\varepsilon^ 2} + \\mathcal{O}\\left(\\frac{1}{n}\\right)\n\\end{equation*}\nconcluding the proof, since $\\delta$ was arbitrary small.\n\\end{proof}\n\\end{lemma}\n\n\\begin{rem}\nNote that the convergence in Lemma \\ref{lawoflarge} actually takes place also in $L^2$. However, to obtain consistency of our estimators, the convergence in probability suffices.\n\\end{rem}\n\n\nAssume that $(X^2_1, X^2_2, \\hdots ,X^2_N)$ is an observed series from an generalized ARCH process $(X_t)_{t\\in\\mathbb{Z}}$. We use the following estimator of the autocovariance function of $X_t^2$\n\n\\begin{equation*}\n\\hat{\\gamma}_N(n) = \\frac{1}{N} \\sum_{t=1}^{N-n} \\left(X^2_t-\\bar{X^2}\\right)\\left(X^2_{t+n}-\\bar{X^2}\\right)\\quad\\text{for }n\\geq 0,\n\\end{equation*}\nwhere $\\bar{X^2}$ is the sample mean of the observations. We show that the estimator above is consistent in two steps. Namely, we consider the sample mean and the term\n\n\\begin{equation*}\n\\frac{1}{N}\\sum_{t=1}^{N-n} X^2_tX_{t+n}^2\n\\end{equation*}\nseparately. If the both terms are consistent, consistency of the autocovariance estimator follows.\n\n\\begin{lemma}\n\\label{lemma:consistency1}\nSuppose $\\mathbb{E}(\\epsilon_0^4) < \\infty$ and $s(t) = cov(L_0L_t) \\to 0$ as $t\\to\\infty$. If $\\alpha_1 < \\frac{1}{\\sqrt{\\mathbb{E}(\\epsilon_0^ 4)}}$, then the sample mean\n\\begin{equation*}\n\\hat{\\mu}_N = \\frac{1}{N}\\sum_{t=1}^N X_t^2\n\\end{equation*}\nconverges in probability to $\\mathbb{E}(X_0^2)$.\n\\begin{proof}\nBy Lemma \\ref{lawoflarge} it suffices to show that $cov(X_1^2, X_{t+1}^2)$ converges to zero as $t$ tends to infinity. For simplicity, let us assume that $t\\geq 2$. Now by fixing $k=t-1$ in \\eqref{iteration} we have\n\n\\begin{equation*}\n\\begin{split}\nX_{t+1}^ 2 &= \\epsilon_{t+1}^2 \\left(\\left(\\prod_{i=0}^{t-1} A_{t-i} \\right) \\sigma_1^2+ \\sum_{i=0}^{t-1}\\left(\\prod_{j=0}^{i-1} A_{t-j}\\right)B_{t-i}\\right)\\\\ \n&= \\epsilon_{t+1}^2 \\left(\\left(\\prod_{i=0}^{t-2} A_{t-i} \\right) \\alpha_1X_1^2 + \\sum_{i=0}^{t-1}\\left(\\prod_{j=0}^{i-1} A_{t-j}\\right)B_{t-i}\\right).\n\\end{split}\n\\end{equation*}\nHence\n\n\\begin{equation*}\n\\begin{split}\nX_{t+1}^2X_1^2=\\left(\\prod_{i=0}^ {t-2} \\alpha_1 \\epsilon_{t-i}^ 2\\right) \\alpha_1 X_1^4 \\epsilon_{t+1}^ 2 + \\epsilon_{t+1}^ 2 X_1^2\\sum_{i=0}^ {t-1}\\left(\\prod_{j=0}^ {i-1} \\alpha_1 \\epsilon_{t-j}^2\\right)(\\alpha_0+l_1L_{t-i}).\n\\end{split}\n\\end{equation*}\nTaking expectations yields\n\n\n\\begin{equation*}\n\\begin{split}\n\\mathbb{E}(X_{t+1}^2X_1^ 2) = \\alpha_1^ t \\mathbb{E}(X_1^ 4) + \\alpha_0\\mathbb{E}(X_1^ 2) \\sum_{i=0}^ {t-1} \\alpha_1^ i + l_1\\sum_{i=0}^ {t-1} \\alpha_1^ i \\mathbb{E}(X_1^2L_{t-i}).\n\\end{split}\n\\end{equation*}\nBy Lemma \\ref{covariance}, and since $\\alpha_1 < 1$ we obtain that\n\n\\begin{footnotesize}\n\\begin{equation*}\n\\begin{split}\n\\mathbb{E}(X_{t+1}^2X_1^ 2) &= \\alpha_1^ t \\mathbb{E}(X_0^ 4) + \\alpha_0\\mathbb{E}(X_0^ 2) \\sum_{i=0}^ {t-1} \\alpha_1^ i +l_1\\sum_{i=0}^{t-1} \\alpha_1^ i\\left(\\frac{\\alpha_0}{1-\\alpha_1 }+ l_1\\sum_{j=0}^ \\infty \\alpha_1^ j f(i-t-j)\\right)\\\\\n&=  \\alpha_1^ t \\mathbb{E}(X_0^ 4) + \\left(\\alpha_0\\mathbb{E}(X_0^ 2) +\\frac{l_1\\alpha_0}{1-\\alpha_1}\\right) \\sum_{i=0}^ {t-1} \\alpha_1^ i + l_1^ 2\\sum_{i=0}^ {t-1} \\sum_{j=0}^ \\infty \\alpha_1^ {i+j} f(i-t-j).\n\\end{split}\n\\end{equation*}\n\\end{footnotesize}\nAs $t$ tends to infinity\n\n\\begin{equation*}\n\\begin{split}\n\\lim_{t\\to\\infty} \\mathbb{E}(X_{t+1}^2X_1^ 2) &= \\frac{\\alpha_0\\mathbb{E}(X_0^ 2)}{1-\\alpha_1} + \\frac{l_1\\alpha_0}{(1-\\alpha_1)^ 2}+l_1^ 2\\lim_{t\\to\\infty}\\sum_{i=0}^ {t-1} \\sum_{j=0}^ \\infty \\alpha_1^ {i+j} f(i-t-j)\\\\\n&= \\frac{\\alpha_0^ 2+2\\alpha_0 l_1}{(1-\\alpha_1)^ 2} + l_1^ 2\\lim_{t\\to\\infty}\\sum_{i=0}^ {\\infty} \\sum_{j=0}^ \\infty \\alpha_1^ {i+j} f(i-t-j),\n\\end{split}\n\\end{equation*}\nwhere we have used \\eqref{meanofX} for expectation of $X_0^2$. Note that $f(t) = s(t) +1$. Hence, there exists $M >0$ such that for the terms  in the double sum it holds that\n\n\\begin{equation*}\n\\left|\\alpha_1^ {i+j} f(i-t-j)\\right| \\leq M \\alpha_1^ {i+j}\\qquad\\text{for every }i, j, t.\n\\end{equation*}\nThus we have a uniform integrable upper bound and consequently, dominated convergence theorem yields\n\n\\begin{equation*}\n\\lim_{t\\to\\infty}\\sum_{i=0}^ {\\infty} \\sum_{j=0}^ \\infty \\alpha_1^ {i+j} f(i-t-j) = \\sum_{i=0}^ \\infty \\sum_{j=0}^ {\\infty} \\alpha_1^ {i+j} = \\frac{1}{(1-\\alpha_1)^ 2}.\n\\end{equation*}\nFinally, we may conclude that\n\n\\begin{equation*}\n\\lim_{t\\to\\infty} \\mathbb{E}(X_{t+1}^2X_1^ 2) = \\left(\\frac{\\alpha_0+l_1}{1-\\alpha_1}\\right)^ 2 = \\mathbb{E}(X_1^ 2)^2.\n\\end{equation*}\n\\end{proof}\n\\end{lemma}\n\n\n\n\\begin{lemma}\n\\label{lemma:consistency2}\nSuppose $\\mathbb{E}(L_0^ 4)<\\infty$ and $\\mathbb{E}(\\epsilon_0^ 8) <\\infty$. In addition, assume that for every fixed $n, n_1$ and $n_2$ it holds that $cov(L_0,L_t)\\to 0$, $cov(L_0L_n, L_{\\pm t}) \\to 0$ and $cov(L_0L_{n_1}, L_tL_{t+n_2})\\to 0$ as $t\\to\\infty$. If $\\alpha_1 < \\frac{1}{\\mathbb{E}(\\epsilon_0^8)^{\\frac{1}{4}}}$, then\n\\begin{equation*}\n\\frac{1}{N-n}\\sum_{t=1}^{N-n} X^2_tX_{t+n}^2\n\\end{equation*}\nconverges in probability to $\\mathbb{E}(X_0^2X_n^2)$ for every $n\\in\\mathbb{Z}$.\n\\begin{proof}\nAgain, by Lemma \\ref{lawoflarge} it suffices to show that $cov(X_0^2X_n^2, X_t^2X_{t+n}^2)$ converges to zero as $t$ tends to infinity. Hence we assume that $t > n$. By \\eqref{uniquesolution} \n\n\\begin{footnotesize}\n\\begin{equation}\n\\label{whatsthelimit}\n\\begin{split}\n\\mathbb{E}(X_0^ 2X_n^ 2X_t^ 2X_{t+n}^ 2) &= \\mathbb{E} \\sum_{i_1=0}^ \\infty  \\sum_{i_2=0}^ \\infty  \\sum_{i_3=0}^ \\infty  \\sum_{i_4=0}^ \\infty \\left(\\prod_{j=0}^ {i_1-1} A_{-1-j}\\right)B_{-1-i_1}\\epsilon_0^ 2 \\left(\\prod_{j=0}^ {i_2-1} A_{n-1-j}\\right)B_{n-1-i_2}\\epsilon_n^ 2\\\\ \n&\\quad \\left(\\prod_{j=0}^ {i_3-1} A_{t-1-j}\\right)B_{t-1-i_3}\\epsilon_t^ 2 \\left(\\prod_{j=0}^ {i_4-1} A_{t+n-1-j}\\right)B_{t+n-1-i_4}\\epsilon_{t+n}^ 2.\n\\end{split}\n\\end{equation}\n\\end{footnotesize}\nSince the summands are non-negative, we can take the expectation inside. Furthermore, by independence of the sequences $\\epsilon_t$ and $L_t$ we observe\n\\begin{equation}\n\\label{whatsthelimit2}\n\\begin{split}\n\\mathbb{E}(X_0^ 2X_n^ 2X_t^ 2X_{t+n}^ 2) &= \\sum_{i_1=0}^ \\infty  \\sum_{i_2=0}^ \\infty  \\sum_{i_3=0}^ \\infty  \\sum_{i_4=0}^ \\infty \\mathbb{E}\\left(B_{-1-i_1} B_{n-1-i_2} B_{t-1-i_3} B_{t+n-1-i_4}\\right)\\\\\n&\\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_n^ 2\\epsilon_t^ 2\\epsilon_{t+n}^ 2\\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{j=0}^ {i_2-1} A_{n-1-j} \\prod_{j=0}^ {i_3-1} A_{t-1-j}\\prod_{j=0}^ {i_4-1} A_{t+n-1-j}\\right).\n\\end{split}\n\\end{equation}\nNext we justify the use of the dominated convergence theorem in order to change the order of the summations and taking the limit. Consequently, it suffices to study the limits of the terms \n\\begin{equation}\n\\label{twoterms}\n\\begin{split}\n&\\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_n^ 2\\epsilon_t^ 2\\epsilon_{t+n}^ 2\\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{j=0}^ {i_2-1} A_{n-1-j} \\prod_{j=0}^ {i_3-1} A_{t-1-j}\\prod_{j=0}^ {i_4-1} A_{t+n-1-j}\\right)\\cdot\\\\\n&\\quad \\mathbb{E}\\left(B_{-1-i_1} B_{n-1-i_2} B_{t-1-i_3} B_{t+n-1-i_4}\\right).\n\\end{split}\n\\end{equation}\n\\textbf{Step 1: finding summable upper bound.}\\\\\nFirst note that the latter term is bounded by a constant. Indeed,\nby stationarity of $(B_t)_{t\\in\\mathbb{Z}}$ we can write\n\\begin{equation}\n\\label{latterterm}\n\\begin{split}\n\\mathbb{E}\\left(B_{-i_1} B_{n-i_2} B_{t-i_3} B_{t+n-i_4}\\right) &= \\alpha_0^ 4 + 4\\alpha_0^ 3l_1 + \\alpha_0^ 2l_1^ 2\\big(\\mathbb{E}(L_{-i_1}L_{n-i_2}) + \\mathbb{E}(L_{-i_1}L_{t-i_3})\\\\\n& + \\mathbb{E}(L_{-i_1}L_{t+n-i_4}) + \\mathbb{E}(L_{n-i_2}L_{t-i_3}) + \\mathbb{E}(L_{n-i_2}L_{t+n-i_4})\\\\\n& + \\mathbb{E}(L_{t-i_3}L_{t+n-i_4})\\big) + \\alpha_0l_1^ 3\\big(\\mathbb{E}(L_{-i_1}L_{n-i_2}L_{t-i_3})\\\\\n& + \\mathbb{E}(L_{-i_1}L_{t-i_3}L_{t+n-i_4}) + \\mathbb{E}(L_{-i_1}L_{n-i_2}L_{t+n-i_4})\\\\ \n&+ \\mathbb{E}(L_{n-i_2}L_{t-i_3}L_{t+n-i_4})\\big) + l_1^ 4\\mathbb{E}(L_{-i_1}L_{n-i_2}L_{t-i_3}L_{t+n-i_4}),\n\\end{split}\n\\end{equation}\nwhich is bounded by a repeated application of Cauchy-Schwarz inequality and the fact that the fourth moment of $L_0$ is finite. \\\\\nConsider now the first term in \\eqref{twoterms}. First we recall the elementary fact\n\\begin{equation}\n\\label{holders}\n1 = \\mathbb{E}(\\epsilon_0^ 2) \\leq \\sqrt{\\mathbb{E}(\\epsilon_0^ 4)} \\leq \\mathbb{E}(\\epsilon_0^ 6)^{\\frac{1}{3}} \\leq \\mathbb{E}(\\epsilon_0^ 8)^{\\frac{1}{4}} < \\infty. \n\\end{equation}\nNext note that the first term in \\eqref{twoterms} is bounded for every set of indices. Indeed, this follows from the independence of $\\epsilon$ and the observation that we obtain terms up to power 8 at most. That is, terms of form $\\epsilon_t^8$ and by assumption, $\\mathbb{E} (\\epsilon_t^8) < \\infty$. Let now $n>0$. Then \n\n\\begin{equation*}\n\\begin{split}\n&\\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_n^ 2\\epsilon_t^ 2\\epsilon_{t+n}^ 2\\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{j=0}^ {i_2-1} A_{n-1-j} \\prod_{j=0}^ {i_3-1} A_{t-1-j}\\prod_{j=0}^ {i_4-1} A_{t+n-1-j}\\right)\\\\\n&= \\mathbb{E}\\left(\\epsilon_{t+n}^2\\right)\\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_n^ 2\\epsilon_t^ 2\\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{j=0}^ {i_2-1} A_{n-1-j} \\prod_{j=0}^ {i_3-1} A_{t-1-j}\\prod_{j=0}^ {i_4-1} A_{t+n-1-j}\\right)\\\\\n&\\leq \\mathbb{E}\\left(\\epsilon_{t+n}^2\\right)\\mathbb{E}\\left(\\epsilon_{t}^4\\right)\\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_n^ 2\\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{j=0}^ {i_2-1} A_{n-1-j} \\prod_{j=0}^ {i_3-1} A_{t-1-j}\\prod_{\\mathclap{\\begin{subarray}{c}\nj=0\\\\ \nj \\neq n-1\n\\end{subarray}}}^ {i_4-1} A_{t+n-1-j}\\right)\\\\\n&\\leq\\mathbb{E}\\left(\\epsilon_{t+n}^2\\right)\\mathbb{E}\\left(\\epsilon_{t}^4\\right)\\mathbb{E}\\left(\\epsilon_{n}^6\\right)\\mathbb{E}\\left(\\epsilon_0^ 2\\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{j=0}^ {i_2-1} A_{n-1-j} \\prod_{\\mathclap{\\begin{subarray}{c}\nj=0\\\\ \nj \\neq t-1-n\n\\end{subarray}}}^ {i_3-1} A_{t-1-j}\\prod_{\\mathclap{\\begin{subarray}{c}\nj=0\\\\ \nj \\neq n-1\\\\\nj\\neq t-1\n\\end{subarray}}}^ {i_4-1} A_{t+n-1-j}\\right)\\\\\n&\\leq\\mathbb{E}\\left(\\epsilon_{t+n}^2\\right)\\mathbb{E}\\left(\\epsilon_{t}^4\\right)\\mathbb{E}\\left(\\epsilon_{n}^6\\right)\\mathbb{E}\\left(\\epsilon_{0}^8\\right)\n\\mathbb{E}\\left(\\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{\\mathclap{\\begin{subarray}{c}\nj=0\\\\ \nj \\neq n-1\n\\end{subarray}}}^ {i_2-1} A_{n-1-j} \\prod_{\\mathclap{\\begin{subarray}{c}\nj=0\\\\ \nj \\neq t-1-n\\\\\nj\\neq t-1\n\\end{subarray}}}^ {i_3-1} A_{t-1-j}\\prod_{\\mathclap{\\begin{subarray}{c}\nj=0\\\\ \nj \\neq n-1\\\\\nj \\neq t-1\\\\\nj \\neq t+n-1\n\\end{subarray}}}^ {i_4-1} A_{t+n-1-j}\\right).\n\\end{split}\n\\end{equation*}\nComputing similarly for $n=0$, using stationarity of $A$,  and observing that \n$$\n1=\\mathbb{E}(\\epsilon_0^ 2) \\leq \\mathbb{E}(\\epsilon_0^ 4) \\leq \\mathbb{E}(\\epsilon_0^ 6) \\leq \\mathbb{E}(\\epsilon_0^ 8) \n$$\nwe hence deduce \n\\begin{equation}\n\\label{loose}\n\\begin{split}\n&\\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_n^ 2\\epsilon_t^ 2\\epsilon_{t+n}^ 2\\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{j=0}^ {i_2-1} A_{n-1-j} \\prod_{j=0}^ {i_3-1} A_{t-1-j}\\prod_{j=0}^ {i_4-1} A_{t+n-1-j}\\right)\\\\\n&\\leq C\\mathbb{E}\\left(\\prod_{j=0}^ {i_1-1} A_{-j} \\prod_{j=0}^ {i_2-1} A_{n-j} \\prod_{j=0}^ {i_3-1} A_{t-j}\\prod_{j=0}^ {i_4-1} A_{t+n-j}\\right),\n\\end{split}\n\\end{equation}\nwhere $C$ is a constant. Moreover, by using similar arguments we observe \n\\begin{footnotesize}\n\\begin{equation*}\n\\mathbb{E}\\left(\\prod_{j=0}^ {i_1-1} A_{-j} \\prod_{j=0}^ {i_2-1} A_{n-j} \\prod_{j=0}^ {i_3-1} A_{t-j}\\prod_{j=0}^ {i_4-1} A_{t+n-j}\\right) \\leq \\mathbb{E}\\left(\\prod_{j=0}^ {i_1-1} A_{-j} \\prod_{j=0}^ {i_2-1} A_{-j} \\prod_{j=0}^ {i_3-1} A_{-j}\\prod_{j=0}^ {i_4-1} A_{-j}\\right).\n\\end{equation*}\n\\end{footnotesize}\nCombining all the estimates above, it thus suffices to prove that \n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1=0}^ \\infty  \\sum_{i_2=0}^ \\infty  \\sum_{i_3=0}^ \\infty  \\sum_{i_4=0}^ \\infty\\mathbb{E}\\left(\\prod_{j=0}^ {i_1-1} A_{-j} \\prod_{j=0}^ {i_2-1} A_{-j} \\prod_{j=0}^ {i_3-1} A_{-j}\\prod_{j=0}^ {i_4-1} A_{-j}\\right)\\\\ \n&\\leq4!  \\sum_{i_4=0}^ \\infty  \\sum_{i_3=0}^{i_4}  \\sum_{i_2=0}^{i_3}  \\sum_{i_1=0}^{i_2}\\mathbb{E}\\left(\\prod_{j=0}^ {i_1-1} A_{-j} \\prod_{j=0}^ {i_2-1} A_{-j} \\prod_{j=0}^ {i_3-1} A_{-j}\\prod_{j=0}^ {i_4-1} A_{-j}\\right) < \\infty.\n\\end{split}\n\\end{equation*}\nNow for $i_1\\leq i_2 \\leq i_3 \\leq i_4$ we have\n\n\\begin{equation*}\n\\mathbb{E}\\left(\\prod_{j=0}^ {i_1-1} A_{-j} \\prod_{j=0}^ {i_2-1} A_{-j} \\prod_{j=0}^ {i_3-1} A_{-j}\\prod_{j=0}^ {i_4-1} A_{-j}\\right) = \\alpha_1^ {i_1+i_2+i_3+i_4} \\mathbb{E}(\\epsilon_0^ 8)^ {i_1}\\mathbb{E}(\\epsilon_0^ 6)^ {i_2-i_1} \\mathbb{E}(\\epsilon_0^4)^{i_3-i_2}\n\\end{equation*}\nwhich yields \n\n\n\\begin{equation*}\n\\begin{split}\n&4!  \\sum_{i_4=0}^ \\infty  \\sum_{i_3=0}^{i_4}  \\sum_{i_2=0}^{i_3}  \\sum_{i_1=0}^{i_2}\\mathbb{E}\\left(\\prod_{j=0}^ {i_1-1} A_{-j} \\prod_{j=0}^ {i_2-1} A_{-j} \\prod_{j=0}^ {i_3-1} A_{-j}\\prod_{j=0}^ {i_4-1} A_{-j}\\right) \\\\\n&= 4!  \\sum_{i_4=0}^ \\infty  \\sum_{i_3=0}^{i_4}  \\sum_{i_2=0}^{i_3}  \\sum_{i_1=0}^{i_2} \\alpha_1^ {i_1+i_2+i_3+i_4} \\mathbb{E}(\\epsilon_0^ 8)^ {i_1}\\mathbb{E}(\\epsilon_0^ 6)^ {i_2-i_1} \\mathbb{E}(\\epsilon_0^4)^{i_3-i_2}\\\\\n&= 4! \\sum_{i_4=0}^\\infty  \\alpha_1^{i_4} \\sum_{i_3=0}^{i_4}  \\left(\\alpha_1 \\mathbb{E}(\\epsilon_0^ 4)\\right)^ {i_3}     \\sum_{i_2=0}^{i_3}     \\left(\\alpha_1  \\frac{\\mathbb{E}(\\epsilon_0^ 6)}{\\mathbb{E}(\\epsilon_0^ 4)}\\right)^ {i_2}     \\sum_{i_1=0}^{i_2} \\left(\\alpha_1 \\frac{\\mathbb{E}(\\epsilon_0^ 8)}{\\mathbb{E}(\\epsilon_0^ 6)}\\right)^{i_1}.\n\\end{split}\n\\end{equation*}\nDenote \n\n\\begin{equation*}\na_1 = \\alpha_1 \\frac{\\mathbb{E}(\\epsilon_0^8)}{\\mathbb{E}(\\epsilon_0^6)}, \\quad a_2 =  \\alpha_1 \\frac{\\mathbb{E}(\\epsilon_0^6)}{\\mathbb{E}(\\epsilon_0^4)} \\quad\\text{and}\\quad a_3 = \\alpha_1 \\mathbb{E}(\\epsilon_0^4). \n\\end{equation*}\nThen we need to show that\n\\begin{equation}\n\\label{quadruple}\n\\begin{split}\n\\sum_{i_4=0}^\\infty  \\alpha_1^{i_4} \\sum_{i_3=0}^{i_4}  a_3^ {i_3}     \\sum_{i_2=0}^{i_3}     a_2^ {i_2}     \\sum_{i_1=0}^{i_2} a_1^{i_1} < \\infty.\n\\end{split}\n\\end{equation} \nFor this suppose first that $1 \\notin S \\coloneqq \\{a_1, a_2, a_3, a_1a_2, a_2a_3, a_1a_2a_3\\}.$ Then we are able to use geometric sums to obtain \n\n\\begin{equation*}\n \\sum_{i_1=0}^{i_2} a_1^{i_1} = \\frac{1-a_1^{i_2+1}}{1-a_1}\\qquad\\text{for }a_1\\neq 1.\n\\end{equation*}\nContinuing like this in the iterated sums in \\eqref{quadruple} we deduce \n\n\\begin{equation*}\n\\begin{split}\n\\sum_{i_2=0}^{i_3} a_2^{i_2} (1-a_1^{i_2+1}) &= \\sum_{i_2=0}^{i_3} a_2^{i_2} - a_1\\sum_{i_2=0}^{i_3} (a_1a_2)^{i_2} = \\frac{1-a_2^{i_3+1}}{1-a_2} - a_1\\frac{1-(a_1a_2)^{i_3+1}}{1-a_1a_2},\n\\end{split}\n\\end{equation*}\n\n\\begin{equation*}\n\\sum_{i_3=0}^{i_4} a_3^{i_3} (1-a_2^{i_3+1}) = \\frac{1-a_3^{i_4+1}}{1-a_3} - a_2\\frac{1-(a_2a_3)^{i_4+1}}{1-a_2a_3},\n\\end{equation*}\nand\n\n\\begin{equation*}\n\\sum_{i_3=0}^{i_4} a_3^{i_3} (1- (a_1a_2)^{i_3+1}) = \\frac{1-a_3^{i_4+1}}{1-a_3} - a_1a_2\\frac{1-(a_1a_2a_3)^{i_4+1}}{1-a_1a_2a_3}.\n\\end{equation*}\nConsequently, it suffices that the following three series converge \n\\begin{equation*}\n\\sum_{i_4=0}^\\infty \\alpha_1^{i_4}a_3^{i_4+1}, \\quad \\sum_{i_4=0}^\\infty \\alpha_1^{i_4} (a_2a_3)^{i_4+1}\\quad\\text{and}\\quad \\sum_{i_4=0}^\\infty \\alpha_1^{i_4} (a_1a_2a_3)^{i_4+1}\n\\end{equation*}\nyielding constraints\n\n\\begin{equation*}\n\\alpha_1 < \\frac{1}{\\sqrt{\\mathbb{E}(\\epsilon_0^4)}}, \\quad \\alpha_1 < \\frac{1}{\\mathbb{E}(\\epsilon_0^6)^{\\frac{1}{3}}} \\quad\\text{and}\\quad \\alpha_1 <\\frac{1}{\\mathbb{E}(\\epsilon_0^8)^{\\frac{1}{4}}}.\n\\end{equation*}\nHowever, these follow from the assumption $\\alpha_1 <\\frac{1}{\\mathbb{E}(\\epsilon_0^8)^{\\frac{1}{4}}}$. Finally, \nif $1 \\in S$ it simply suffices to replace $a_1,a_2, a_3$ with\n\\begin{equation*}\n\\tilde{a}_1 = \\alpha_1 \\left(\\frac{\\mathbb{E}(\\epsilon_0^8)}{\\mathbb{E}(\\epsilon_0^6)}+\\delta\\right), \\quad \\tilde{a}_2 =  \\alpha_1 \\left(\\frac{\\mathbb{E}(\\epsilon_0^6)}{\\mathbb{E}(\\epsilon_0^4)}+\\delta\\right) \\quad\\text{and}\\quad \\tilde{a}_3 = \\alpha_1 \\left(\\mathbb{E}(\\epsilon_0^4)+\\delta\\right) \n\\end{equation*}\nsuch that \n$$\n1 \\notin \\{\\tilde{a}_1, \\tilde{a}_2, \\tilde{a}_3, \\tilde{a}_1\\tilde{a}_2, \\tilde{a}_2\\tilde{a}_3, \\tilde{a}_1\\tilde{a}_2\\tilde{a}_3\\}.\n$$\nChoosing $\\delta<0$ small enough the claim follows from the fact that the inequality $\\alpha_1 <\\frac{1}{\\mathbb{E}(\\epsilon_0^8)^{\\frac{1}{4}}}$ is strict.\\\\\n\\textbf{Step 2: computing the limit of \\eqref{whatsthelimit}.}\\\\\nBy step 1 we can apply dominated convergence theorem in \\eqref{whatsthelimit}. For this let us analyze the limit behaviour of \\eqref{twoterms}. For the latter term we use \\eqref{latterterm}. By assumptions, we have e.g. the following identities:\n\n\\begin{align*}\n\\lim_{t\\to\\infty} \\mathbb{E}(L_{t-i_3}L_{t+n-i_4}) &=1\\\\ \n\\lim_{t\\to\\infty} \\mathbb{E}(L_{-i_1} L_{t-i_3}L_{t+n-i_4}) &= f(n+i_3-i_4)\\\\\n\\lim_{t\\to\\infty} \\mathbb{E}(L_{-i_1}L_{n-i_2}L_{t-i_3}L_{t+n-i_4}) &= f(n+i_1-i_2)f(n+i_3-i_4).\n\\end{align*} \nTherefore the limit of the latter term of \\eqref{twoterms} is given by\n\n\\begin{scriptsize}\n\\begin{equation*}\n\\begin{split}\n\\lim_{t\\to\\infty}\\mathbb{E}\\left(B_{-i_1} B_{n-i_2} B_{t-i_3} B_{t+n-i_4}\\right) &= \\alpha_0^ 4 + 4\\alpha_0^ 3l_1 + \\alpha_0^ 2l_1^ 2\\big(4+ f(n+i_1-i_2)+f(n+i_3-i_4)\\big)\\\\\n& +\\alpha_0l_1^ 3\\big(f(n+i_1-i_2) + f(n+i_3-i_4) + f(n+i_1-i_2)\\\\\n&+ f(n+i_3-i_4)\\big) +l_1^4f(n+i_1-i_2)f(n+i_3-i_4)\\\\\n&=  (\\alpha_0^ 2 + 2\\alpha_0l_1+l_1^ 2f(n+i_1-i_2))(\\alpha_0^2+2\\alpha_0l_1+l_1^ 2f(n+i_3-i_4))\n\\end{split}\n\\end{equation*}\n\\end{scriptsize}\nThe first term of \\eqref{twoterms} can be divided into two independent parts whenever $t$ is large enough. More precisely, for $t > \\max \\{n+i_3, i_4\\}$, we have\n\n\\begin{equation*}\n\\begin{split}\n&\\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_n^ 2\\epsilon_t^ 2\\epsilon_{t+n}^ 2\\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{j=0}^ {i_2-1} A_{n-1-j} \\prod_{j=0}^ {i_3-1} A_{t-1-j}\\prod_{j=0}^ {i_4-1} A_{t+n-1-j}\\right)\\\\ \n=&  \\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_n^ 2 \\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{j=0}^ {i_2-1} A_{n-1-j}\\right)\\mathbb{E}\\left(\\epsilon_t^ 2\\epsilon_{t+n}^ 2 \\prod_{j=0}^ {i_3-1} A_{t-1-j} \\prod_{j=0}^ {i_4-1} A_{t+n-1-j}\\right)\\\\ \n=& \\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_n^ 2 \\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{j=0}^ {i_2-1} A_{n-1-j}\\right)\\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_{n}^ 2 \\prod_{j=0}^ {i_3-1} A_{-1-j} \\prod_{j=0}^ {i_4-1} A_{n-1-j}\\right),\n\\end{split}\n\\end{equation*}\nwhere the last equality follows from stationarity of $A_t$. Hence\n\n\\begin{footnotesize}\n\\begin{equation*}\n\\begin{split}\n&\\lim_{t\\to\\infty} \\mathbb{E}(X_0^ 2X_n^ 2X_t^ 2X_{t+n}^ 2)\\\\\n =& \\sum_{i_1=0}^ \\infty  \\sum_{i_2=0}^ \\infty  \\sum_{i_3=0}^ \\infty  \\sum_{i_4=0}^ \\infty \\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_n^ 2 \\prod_{j=0}^ {i_1-1} A_{-1-j} \\prod_{j=0}^ {i_2-1} A_{n-1-j}\\right)\\mathbb{E}\\left(\\epsilon_0^ 2\\epsilon_{n}^ 2 \\prod_{j=0}^ {i_3-1} A_{-1-j} \\prod_{j=0}^ {i_4-1} A_{n-1-j}\\right)\\cdot\\\\\n& (\\alpha_0^ 2 + 2\\alpha_0l_1+l_1^ 2f(n+i_1-i_2))(\\alpha_0^2+2\\alpha_0l_1+l_1^ 2f(n+i_3-i_4)).\n\\end{split}\n\\end{equation*}\n\\end{footnotesize}\nOn the other hand, by \\eqref{uniquesolution}\n\n\\begin{footnotesize}\n\\begin{equation*}\n\\begin{split}\n\\mathbb{E}(X_0^ 2X_n^ 2) &= \\sum_{i_1 = 0}^\\infty \\sum_{i_2=0}^ \\infty \\mathbb{E}\\left(\\epsilon_0^ 2 \\epsilon_n^ 2\\prod_{j=0}^ {i_1-1}A_{-1-j} \\prod_{j=0}^{i_2-1} A_{n-1-j}\\right) \\mathbb{E}\\left((\\alpha_0+l_1L_{-1-i_1})(\\alpha_0 + l_1L_{n-1-i_2})\\right)\\\\\n&= \\sum_{i_1 = 0}^\\infty \\sum_{i_2=0}^ \\infty \\mathbb{E}\\left(\\epsilon_0^ 2 \\epsilon_n^ 2\\prod_{j=0}^ {i_1-1}A_{-1-j} \\prod_{j=0}^{i_2-1} A_{n-1-j}\\right) (\\alpha_0^ 2 + 2\\alpha_0l_1 + l_1^ 2f(n+i_1-i_2)).\n\\end{split}\n\\end{equation*}\n\\end{footnotesize}\nConsequently, we conclude that \n\n\\begin{equation*}\n\\lim_{t\\to\\infty} \\mathbb{E}(X_0^ 2X_n^ 2X_t^ 2X_{t+n}^ 2) = \\mathbb{E}(X_0^2 X_n^ 2)^ 2\n\\end{equation*}\nproving the claim.\n\\end{proof}\n\\end{lemma}\n\n\n\n\\begin{rem}\nThe assumptions of Lemma \\ref{lemma:consistency2} cohere with the assumptions of Lemma \\ref{lemma:consistency1}. Moreover, the assumptions made related to convergence of covariances are very natural. Indeed, we only assume that the (linear) dependencies within the process $L_t$ vanish over time. Examples of $L$ satisfying the required assumptions can be found in Section \\ref{examples}.\n\\end{rem}\n\n\n\n\n\n\\subsection{Estimation of the model parameters}\nSet, for $N\\geq 1$,\n\n\\begin{equation*}\n\\hat{\\mu}_{2,N} = \\frac{1}{N}\\sum_{t=1}^{N} X^4_t\n\\end{equation*}\nand\n\n\\begin{equation*}\ng_0({\\pmb{\\gamma}}_0) = b_0({\\pmb{\\gamma}}_0)^2 - 4a_0({\\pmb{\\gamma}}_0)c_0({\\pmb{\\gamma}}_0),\n\\end{equation*}\nwhere $a_0({\\pmb{\\gamma}}_0)$, $b_0({\\pmb{\\gamma}}_0)$ and $c_0({\\pmb{\\gamma}}_0)$ are as in \\eqref{quadraticterms3}. In addition, let \n\\begin{equation*}\n{\\hat{\\pmb{\\gamma}}_{0,N}} = [\\hat{\\gamma}_N(n+1), \\hat{\\gamma}_N(n), \\hat{\\gamma}_N(n-1), \\hat{\\gamma}_N(1), \\hat{\\gamma}_N(0), \\hat{\\mu}_{2,N}]\n\\end{equation*}\nand $\\hat{{\\pmb{\\xi}}}_{0, N} = [{\\hat{\\pmb{\\gamma}}_{0,N}}, \\hat{\\mu}_N]$ for some fixed $n\\neq 0$. The following estimators are motivated by \\eqref{alphaofgamma2}, \\eqref{lofgamma2} and \\eqref{alpha0ofgamma2}. \n\n\\begin{defi}\\label{def1}\nWe define estimators $\\hat{\\alpha}_1$, $\\hat{l}_1$ and $\\hat{\\alpha}_0$ for the model parameters $\\alpha_1$, $l_1$ and $\\alpha_0$ respectively through\n\n\\begin{equation}\n\\label{haaalpha1}\n\\hat{\\alpha}_1 = \\alpha_1({\\hat{\\pmb{\\gamma}}_{0,N}})=  \\frac{-b_0({\\hat{\\pmb{\\gamma}}_{0,N}}) \\pm \\sqrt{g_0({\\hat{\\pmb{\\gamma}}_{0,N}})}}{2a_0({\\hat{\\pmb{\\gamma}}_{0,N}})},\n\\end{equation}\n\n\\begin{footnotesize}\n\\begin{equation}\n\\label{haal1}\n\\hat{l}_1=l_1({\\hat{\\pmb{\\gamma}}_{0,N}}) = \\sqrt{\\frac{1}{s(0)}\\left(\\alpha_1({\\hat{\\pmb{\\gamma}}_{0,N}})^2 \\hat{\\gamma}_N(0) - 2\\alpha_1({\\hat{\\pmb{\\gamma}}_{0,N}}) \\hat{\\gamma}_N(1)+ \\hat{\\gamma}_N(0) - \\frac{\\hat{\\mu}_{2,N} Var(\\epsilon_0^2)}{\\mathbb{E}(\\epsilon_0^4)} \\right)}\n\\end{equation}\n\\end{footnotesize}\nand\n\n\\begin{equation}\n\\label{haaalpha0}\n\\hat{\\alpha}_0 = \\alpha_0(\\hat{{\\pmb{\\xi}}}_{0, N}) = \\hat{\\mu}_N(1-\\alpha_1({\\hat{\\pmb{\\gamma}}_{0,N}})) - l_1({\\hat{\\pmb{\\gamma}}_{0,N}}),\n\\end{equation}\nwhere $n \\neq 0$.\n\n\\end{defi}\n\n\\begin{theorem}\n\\label{theo:consistency}\nAssume that $a_0({\\pmb{\\gamma}}_0) \\neq 0$ and $g_0({\\pmb{\\gamma}}_0) > 0$. Let the assumptions of Lemma \\ref{lemma:consistency2} prevail. Then $\\hat{\\alpha}_1, \\hat{l}_1$ and $\\hat{\\alpha}_0$ given by \\eqref{haaalpha1}, \\eqref{haal1} and \\eqref{haaalpha0} are consistent.\n\\end{theorem}\n\\begin{proof}\nSince the assumptions of Lemma \\ref{lemma:consistency2} are satisfied, so are the assumptions of Lemma \\ref{lemma:consistency1} implying that the autocovariance estimators, the mean and the second moment estimator of $X_t^2$ are consistent. The claim follows from the continuous mapping theorem.\n\\end{proof}\n\n\nLet us denote\n\n\\begin{equation*}\ng({\\pmb{\\gamma}}) = b({\\pmb{\\gamma}})^2 - 4a({\\pmb{\\gamma}})^2,\n\\end{equation*}\nwhere $a({\\pmb{\\gamma}})$ and $b({\\pmb{\\gamma}})$ are as in \\eqref{quadraticterms2}. In addition, let \n\\begin{equation*}\n{\\hat{\\pmb{\\gamma}}_N} = [\\hat{\\gamma}_N(n_1+1), \\hat{\\gamma}_N(n_2+1), \\hat{\\gamma}_N(n_1), \\hat{\\gamma}_N(n_2), \\hat{\\gamma}_N(n_1-1), \\hat{\\gamma}_N(n_2-1)]\n\\end{equation*}\nand $\\hat{{\\pmb{\\xi}}}_N = [{\\hat{\\pmb{\\gamma}}_N}, \\hat{\\mu}_N]$ for some fixed $n_1,n_2\\neq 0$ with $n_1 \\neq n_2$. \nThe following estimators are motivated by \\eqref{alphaofgamma}, \\eqref{lofgamma} and \\eqref{alpha0ofgamma}. \n\n\\begin{defi}\\label{def2}\nWe define estimators $\\hat{\\alpha}_1$, $\\hat{l}_1$ and $\\hat{\\alpha}_0$ for the model parameters $\\alpha_1$, $l_1$ and $\\alpha_0$ respectively through\n\n\\begin{equation}\n\\label{haalpha1}\n\\hat{\\alpha}_1 = \\alpha_1({\\hat{\\pmb{\\gamma}}_N})=  \\frac{-b({\\hat{\\pmb{\\gamma}}_N}) \\pm \\sqrt{g({\\hat{\\pmb{\\gamma}}_N})}}{2a({\\hat{\\pmb{\\gamma}}_N})},\n\\end{equation}\n\n\n\\begin{footnotesize}\n\\begin{equation}\n\\label{hal1}\n\\hat{l}_1=l_1({\\hat{\\pmb{\\gamma}}_N}) = \\sqrt{\\frac{\\alpha_1^2({\\hat{\\pmb{\\gamma}}_N})\\hat{\\gamma}_N(n_2) - \\alpha_1({\\hat{\\pmb{\\gamma}}_N})(\\hat{\\gamma}_N(n_2+1) + \\hat{\\gamma}_N(n_2-1)) + \\hat{\\gamma}_N(n_2)}{s(n_2)}}\n\\end{equation}\n\\end{footnotesize}\nand\n\n\\begin{equation}\n\\label{haalpha0}\n\\hat{\\alpha}_0 = \\alpha_0(\\hat{{\\pmb{\\xi}}}_N) = \\hat{\\mu}_N(1-\\alpha_1({\\hat{\\pmb{\\gamma}}_N})) - l_1({\\hat{\\pmb{\\gamma}}_N}),\n\\end{equation}\nwhere $n_1, n_2\\neq 0$ and $n_1\\neq n_2$.\n\n\\end{defi}\n\n\n\\begin{theorem}\n\\label{theo:consistency2}\nAssume that $s(n_2) \\neq 0, a({\\pmb{\\gamma}}) \\neq 0$ and $g({\\pmb{\\gamma}}) > 0$. Let the assumptions of Lemma \\ref{lemma:consistency2} prevail. Then $\\hat{\\alpha}_1, \\hat{l}_1$ and $\\hat{\\alpha}_0$ given by \\eqref{haalpha1}, \\eqref{hal1} and \\eqref{haalpha0} are consistent.\n\\end{theorem}\n\\begin{proof}\nThe proof is basically the same as with Theorem \\ref{theo:consistency}.\n\\end{proof}\n\n\\begin{rem}\n\\begin{itemize}\n\\item Statements of Theorems \\ref{theo:consistency} and \\ref{theo:consistency2} hold true also when $g_0({\\pmb{\\gamma}}_0) = 0$ and $g({\\pmb{\\gamma}}) = 0$, but in these cases the estimators do not necessarily become real valued as the sample size grows. In comparison, in \\cite{vouti} the estimators were forced to be real by using indicator functions. \n\n\n\\item The estimators from Definitions \\ref{def1} and \\ref{def2} are of course related. In practice (see the next section) we use those from Definition \\ref{def1} while those from Definition \\ref{def2} are needed just in case when we need  more information in order to choose the correct sign for $\\hat\\alpha _1$, see Remark \\ref{rem:sign}.\n\n\\item Note that here we implicitly assumed that the correct sign can be chosen in $\\hat\\alpha_1$. However, this is not a restriction as discussed.\n\\end{itemize}\n\\end{rem}\n\n\n\n\\subsection{Examples}\n\\label{examples}\nWe will present several examples of stationary processes for which  our main result stated in Theorem \\ref{theo:consistency} apply. Our examples are constructed as \n$$L_{t}:= \\left( X_{t+1}- X_{t}\\right) ^{2}, \\mbox{ for every } t\\in \\mathbb{Z} $$\nwhere  $(X_{t})_{t\\in \\mathbb{R}}$  is a stochastic process with stationary increments. We discuss below the case when $X$ is a continuous Gaussian process (the fractional Brownian motion), a continuous non-Gaussian process (the Rosenblatt process), or a jump process (the compensated Poisson process). \n\n\\subsubsection{The fractional Brownian motion}\n\nLet $X_{t}: = B ^{H}_{t}$ for every $t\\in \\mathbb{R} $ where $(B ^{H}_{t}) _{t\\in \\mathbb{R}} $ is a two-sided fractional Brownian motion with Hurst parameter $H\\in (0,1)$. Recall that $ B^{H} $ is a centered Gaussian process with covariance\n\\begin{equation*}\n\\mathbb{E} (B_{t}B_{s})=\\frac{1}{2} (\\vert t\\vert ^{2H}+ \\vert s\\vert^{2H} -\\vert t-s\\vert ^{2H} ), \\hskip0.3cm s,t \\in \\mathbb{R}.\n\\end{equation*}\nLet us verify that the conditions from Lemma \\ref{lemma:consistency2} and Theorem \\ref{theo:consistency} are satisfied by $L_{t}= (B ^{H}_{t+1}- B ^{H}_{t}) ^{2}$. First, notice that (see Lemma 2 in \\cite{BTT}) that for $t\\geq 1$  \n\\begin{equation*}\nCov( L_{0}, L_{t})=\\mathbb{E}\\left( (B^{H}_{1} ) ^{2} (B ^{H}_{t+1} -B ^{H}_{t} ) ^{2}  \\right) -1= 2(r_{H}(t))^2 \n\\end{equation*} \nwith\n\\begin{equation}\n\\label{rh}\nr_{H}(t)= \\frac{1}{2} \\left[ (t+1) ^{2H}+(t-1) ^{2H}-2t ^{2H}\\right]\\to _{t\\to \\infty}0\n\\end{equation}\nsince $r_{H}(t) $ behaves as $t^{2H-2}$ for $t$ large.\n\nLet us now turn to the third-order condition, i.e. $Cov (L_{0}L_{n}, L_{t})= \\mathbb{E} (L_{0} L_{n} L_{t} )-\\mathbb{E} (L_{0} L_{n}  ) \\to 0$ as $t\\to \\infty$. We can suppose $n\\geq 1$ is fixed and $t>n$. \n\nFor any three centered Gaussian random variables $X_{1}, X_{2}, X_{3}$  with unit variance we have $\\mathbb{E} ( X_{1} ^{2} X_{2} ^{2}) = 1+2(\\mathbb{E}(X_{1}X_{2}) ) ^{2} $ and \n\\begin{eqnarray*}\n\\mathbb{E} ( X_{1} ^{2} X_{2} ^{2}X_{3} ^{2})&=& 2 \\left( (\\mathbb{E} (X_{1} X_{2}))^{2} +  (\\mathbb{E} (X_{1} X_{3}))^{2} +(\\mathbb{E} (X_{2} X_{3}))^{2} \\right)\\\\\n&+& 4 \\mathbb{E} (X_{1} X_{2}))\\mathbb{E} (X_{1} X_{3}))\\mathbb{E} (X_{2} X_{3}))+1\\\\\n&=&\\mathbb{E} ( X_{1} ^{2} X_{2} ^{2}) + 2 \\left(   (\\mathbb{E} (X_{1} X_{3}))^{2} +(\\mathbb{E} (X_{2} X_{3}))^{2} \\right) \\\\\n&+& 4 \\mathbb{E} (X_{1} X_{2}))\\mathbb{E} (X_{1} X_{3}))\\mathbb{E} (X_{2} X_{3})).\n\\end{eqnarray*}\nBy applying this formula to $X_{1}= B ^{H}_{1}, X_{2}= B^H_{n+1}- B ^{H}_{n}, X_{3}= B ^{H}_{t+1} -B ^{H}_{t}$, we find\n \\begin{equation*}\nCov (L_{0}L_{n}, L_{t})=2 r_{H}(t) ^{2} + 2r_{H}(t-n) ^{2}+4r_{H}(n)r_{H}(t) r_{H}(t-n)  \n\\end{equation*}\nwhere $r_{H}$ is given by (\\ref{rh}). By (\\ref{rh}), the above expression converges to zero as $t\\to \\infty$.\n\nSimilarly for the fourth-order condition, the formulas are more complex but we can verify by standard calculations that, for every $n_{1}, n_{2}\\geq 1$ and for every $t> \\max (n_{1}, n_{2})$, the quantity \n$$ \\mathbb{E}(L_{0} L_{n_{1}}L_{t}L_{t+n_{2}}) - \\mathbb{E}(L_{0} L_{n_{1}}) \\mathbb{E} (L_{t}L_{t+n_{2}}) $$\ncan be expressed as a polynomial (without term of degree zero) in $r_{H}(t), r_{H}(t-n_{1}), r_{H}(t+n_{2}), r_{H}(t+n_{2} -n_{1}) $ with coefficients depending on $n_{1}, n_{2}$. The conclusion is obtained by (\\ref{rh}). \n\n\n\\subsubsection{The compensated Poisson process}\n\nLet $(N_{t})_{t\\in \\mathbb{R} }$ be a Poisson process with intensity $\\lambda =1$. Recall that $N$ is a cadlag adapted stochastic process, with independent increments, such that for every $s<t$, the random variable $N_{t}-N_{s}$ follows a Poisson distribution with parameter $t-s$. Define the compensated Poisson process $ (\\tilde{N}_{t}) _{t\\in \\mathbb{R} } $ by $\\tilde{N}_{t}=N_{t}-t$ for every $t\\in \\mathbb{R}$ and let $L_{t}= (N_{t+1}-N_{t})^ {2}$.  Clearly $\\mathbb{E}L_{t}= 1$ for every $t$ and, by the independence of the increments of $\\tilde{N}$, we have that for $t$ large enough \n$$Cov(L_{0}, L_{t})= Cov( L_{0}L_{n}, L_{t}) = Cov (L_{0}L_{n_{1}}, L_{t}L_{t+n_{2}})=0, $$ so the conditions in Theorem \\ref{theo:consistency} are fulfilled. \n\n\n\\subsubsection{The Rosenblatt process} \nThe (one-sided) Rosenblatt process $(Z^{H}_{t}) _{t\\geq 0} $ is a self-similar stochastic process with stationary increments  and long memory  in the second Wiener chaos, i.e. it can be expressed as a multiple stochastic integral of order two with respect to the Wiener process. The Hurst parameter $H$ belongs to $(\\frac{1}{2}, 1)$ and it characterizes the main properties of the process.  Its representation is \n$$ Z ^{H}_{t}=  \\int_{\\mathbb{R} }\\int_{\\mathbb{R}} f_{H}(y_{1}, y_{2}) dW(y_{1})dW(y_{2}) $$\n where $(W(y))_{y\\in \\mathbb{R}}$ is Wiener process and $f_{H} $ is  deterministic function such that\\\\ $\\int_{\\mathbb{R}} \\int_{\\mathbb{R}} f_{H}(y_{1}, y_{2}) ^{2} dy_{1}dy_{2} <\\infty$. See e.g. \\cite{T} for a more complete exposition on the Rosenblatt process. The two-sided Rosenblatt process has been introduced in \\cite{Cou}. In particular, it has the same covariance as the fractional Brownian motion, so $\\mathbb{E}(L_{t})= \\mathbb{E}(Z^{H}_{t+1} -Z^{H}_{t}) ^{2}=1$ for every $t$. The use of the Rosenblatt process can be motivated by the presence of the long-memory in the emprical data for liquidity in financial markets, see \\cite{Tsuji}.\n\nThe computation of the quantities $Cov(L_{0}, L_{t}), Cov (L_{0}L_{n}, L_{t})$ and $ Cov (L_{0} L_{n_{1}}, L_{t} L_{t+n_{2}})$  requires rather technical tools from stochastic analysis including properties of multiple integrals and product formula which we prefer to avoid here. We only mention that the term $Cov(L_{0}, L_{t})$ can be written as $P( r_{H}(t), r_{H,1}(t) ) $ where $P$ is a polynomial without term of degree zero, $r_{H}$ is given by (\\ref{rh}), while \n$$r_{H,1}(t)= \\int_{0} ^{1} \\int_{0}^{1} \\int_{t}^{t+1} \\int_{t} ^{t+1} du_{1}du_{2}du_{3}du_{4} \\vert u_{1}-u_{2}\\vert ^{H-1}\\vert u_{2}-u_{3} \\vert ^{H-1}\\vert u_{3}-u_{4}\\vert ^{H-1}\\vert u_{4}-u_{1}\\vert ^{H-1}.$$\nNote that\n$$r_{H,1}(t) = \\int_{[0,1] ^{4}} du_{1}du_{2}du_{3}du_{4} \\vert u_{1}-u_{2}\\vert ^{H-1}\\vert u_{2}-u_{3}+t \\vert ^{H-1}\\vert u_{3}-u_{4}\\vert ^{H-1}\\vert u_{4}-u_{1}+t\\vert ^{H-1}.  $$ \nSince $\\vert u_{1}-u_{2}\\vert ^{H-1}\\vert u_{2}-u_{3}+t \\vert ^{H-1}\\vert u_{3}-u_{4}\\vert ^{H-1}\\vert u_{4}-u_{1}+t\\vert ^{H-1}$ converges to zero as $t\\to \\infty$ for every $u_{i}$ and since  this  integrand is bounded for $t$ large by $\\vert u_{1}-u_{2}\\vert ^{H-1}\\vert u_{2}-u_{3} \\vert ^{H-1}\\vert u_{3}-u_{4}\\vert ^{H-1}\\vert u_{4}-u_{1}\\vert ^{H-1}$, which is integrable over $[0,1] ^{4}$, we obtain, via the dominated convergence theorem, that $Cov(L_{0}, L_{t})\\to _{t\\to \\infty} 0.$\nSimilarly, the quantities $Cov (L_{0}L_{n}, L_{t})$ and $ Cov (L_{0} L_{n_{1}}, L_{t} L_{t+n_{2}})$ can be also expressed as polynomials (without constant terms) of $r_{H}, r_{H, k}$, $k=1,2,3, 4$ where $$r_{H,k}(t) = \\int_{A_{1}\\times ...A_{2k}}  du_{1}...du_{2k}  \\vert u_{1}- u_{2}\\vert ^{H-1}...\\vert u_{2k-1}-u_{2k}\\vert ^{H-1} \\vert u_{2k}-u_{1}\\vert ^{H-1}, $$ where at least one set $A_{i}$ is $(t, t+1)$. Thus we may apply a similar argument as above. \n\n\n\n\n\n\\section{Simulations}\nThis section provides some visual illustrations of convergence of the estimators \\eqref{haaalpha1}, \\eqref{haal1} and \\eqref{haaalpha0} with respect to different liquidities $(L_t)_{t\\in\\mathbb{Z}}$. \\\\\nThe general setting throughout the simulations is the following. The IID process $(\\epsilon_t)_{t\\in\\mathbb{Z}}$ is assumed to be a sequence of standard normals. In this case the restriction given by Lemma \\ref{lemma:consistency2} reads $\\alpha_1 < \\frac{1}{105^{\\frac{1}{4}}} \\approx 0.31$. The lag used is $n=1$ and the true values of the model parameters are $\\alpha_0=1$, $\\alpha_1 = 0.1$ and $l_1 = 0.5$. The used sample sizes are $N=100, N=1000, N=10000$ and $N=100000$. The initial $X_0^2$ is set to a value $1.7$. After the processes $L_t$ with $t=0, 1,...N-2$ and $\\epsilon_t$ with $t=1,2,...N-1$ are simulated, the initial is used to generate $\\sigma_1^2$ using \\eqref{arch}. Together with $\\epsilon_1$ this gives $X_1^2$, after which \\eqref{arch} yields the sample $\\{X_0^2, X_1^2,..., X^2_{N-1}\\}$.\\\\\nIn the first three subsections simulation results of the generalized ARCH process with liquidity given by $L_t = (B^H_{t+1} - B^H_{t})^2$ are presented. The used Hurst indices are $H=\\frac{1}{3}, H=\\frac{2}{3}$ and $H=\\frac{4}{5}$. In the fourth subsection the liquidity process is given by $L_t = (\\tilde{N}_{t+1} - \\tilde{N}_t)^2$, where $N_t$ is a compensated Poisson process with $\\lambda = 1$. \n\nIn all subsections the sample size $N$ is varied, and each setting is repeated $1000$ times to provide histograms of the estimates. Our simulations show that the behaviour of the limit distributions is close to Gaussian one, as $N$ increases. We also note that, since the estimators involve square roots, they may produce complex valued estimates. However, asymptotically the estimates become real. Throughout the simulations the complex valued estimates have been simply removed, although the percentage of complex values is computed in each setting. Finally, some illustrative tables are given in Appendix \\ref{sec:A_tables}.\n\n\\subsection{Fractional Brownian motion with $H=\\frac{1}{3}$.}\nHistograms of the estimates of the model parameters corresponding to $L_t = (B^H_{t+1} - B^H_{t})^2$ with $H = \\frac{1}{3}$ are provided in Figures \\ref{fig:H=1per3-N=100}, \\ref{fig:H=1per3-N=1000}, \\ref{fig:H=1per3-N=10000} and \\ref{fig:H=1per3-N=100000}. The used sample sizes were $N=100, N=1000, N=10000$ and $N= 100000$. The sample sizes $N=100$ and $N=1000$ resulted complex valued estimates in $44.2\\%$ and $3.5\\%$ of the simulations respectively, whereas with the larger sample sizes all the estimates were real.\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=1per3-N=100-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=1per3-N=100-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=1per3-N=100-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=1per3-N=100-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=1per3-N=100-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=1per3-N=100-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{1}{3}$ and $N=100$.}\n\\label{fig:H=1per3-N=100}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=1per3-N=1000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=1per3-N=1000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=1per3-N=1000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=1per3-N=1000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=1per3-N=1000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=1per3-N=1000-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{1}{3}$ and $N=1000$.}\n\\label{fig:H=1per3-N=1000}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=1per3-N=10000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=1per3-N=10000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=1per3-N=10000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=1per3-N=10000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=1per3-N=10000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=1per3-N=10000-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{1}{3}$ and $N=10000$.}\n\\label{fig:H=1per3-N=10000}\n\\end{figure}\n\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=1per3-N=100000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=1per3-N=100000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=1per3-N=100000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=1per3-N=100000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=1per3-N=100000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=1per3-N=100000-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{1}{3}$ and $N=100000$.}\n\\label{fig:H=1per3-N=100000}\n\\end{figure}\n\n\n\\subsection{Fractional Brownian motion with $H=\\frac{2}{3}$.}\nHistograms of the estimates of the model parameters corresponding to $L_t = (B^H_{t+1} - B^H_{t})^2$ with $H = \\frac{2}{3}$ are provided in Figures \\ref{fig:H=2per3-N=100}, \\ref{fig:H=2per3-N=1000}, \\ref{fig:H=2per3-N=10000} and \\ref{fig:H=2per3-N=100000}. The used sample sizes were $N=100, N=1000, N=10000$ and $N= 100000$. The sample sizes $N=100$ and $N=1000$ resulted complex valued estimates in $45.5\\%$ and $2.9\\%$ of the simulations respectively, whereas with the larger sample sizes all the estimates were real.\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=2per3-N=100-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=2per3-N=100-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=2per3-N=100-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=2per3-N=100-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=2per3-N=100-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=2per3-N=100-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{2}{3}$ and $N=100$.}\n\\label{fig:H=2per3-N=100}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=2per3-N=1000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=2per3-N=1000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=2per3-N=1000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=2per3-N=1000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=2per3-N=1000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=2per3-N=1000-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{2}{3}$ and $N=1000$.}\n\\label{fig:H=2per3-N=1000}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=2per3-N=10000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=2per3-N=10000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=2per3-N=10000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=2per3-N=10000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=2per3-N=10000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=2per3-N=10000-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{2}{3}$ and $N=10000$.}\n\\label{fig:H=2per3-N=10000}\n\\end{figure}\n\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=2per3-N=100000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=2per3-N=100000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=2per3-N=100000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=2per3-N=100000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=2per3-N=100000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=2per3-N=100000-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{2}{3}$ and $N=100000$.}\n\\label{fig:H=2per3-N=100000}\n\\end{figure}\n\n\n\n\\subsection{Fractional Brownian motion with $H=\\frac{4}{5}$.}\nHistograms of the estimates of the model parameters corresponding to $L_t = (B^H_{t+1} - B^H_{t})^2$ with $H = \\frac{4}{5}$ are provided in Figures \\ref{fig:H=4per5-N=100}, \\ref{fig:H=4per5-N=1000}, \\ref{fig:H=4per5-N=10000} and \\ref{fig:H=4per5-N=100000}. The used sample sizes were $N=100, N=1000, N=10000$ and $N= 100000$. The sample sizes $N=100$ and $N=1000$ resulted complex valued estimates in $47.9\\%$ and $4.3\\%$ of the simulations respectively, whereas with the larger sample sizes all the estimates were real.\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=4per5-N=100-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=4per5-N=100-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=4per5-N=100-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=4per5-N=100-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=4per5-N=100-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=4per5-N=100-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{4}{5}$ and $N=100$.}\n\\label{fig:H=4per5-N=100}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=4per5-N=1000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=4per5-N=1000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=4per5-N=1000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=4per5-N=1000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=4per5-N=1000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=4per5-N=1000-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{4}{5}$ and $N=1000$.}\n\\label{fig:H=4per5-N=1000}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=4per5-N=10000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=4per5-N=10000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=4per5-N=10000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=4per5-N=10000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=4per5-N=10000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=4per5-N=10000-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{4}{5}$ and $N=10000$.}\n\\label{fig:H=4per5-N=10000}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{H=4per5-N=100000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:H=4per5-N=100000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=4per5-N=100000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:H=4per5-N=100000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{H=4per5-N=100000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:H=4per5-N=100000-l1}\n \\end{subfigure}\n\\caption{Fractional Brownian motion liquidity with $H=\\frac{4}{5}$ and $N=100000$.}\n\\label{fig:H=4per5-N=100000}\n\\end{figure}\n\n\n\\subsection{Compensated Poisson with $\\lambda = 1$.} \nHistograms of the estimates of the model parameters corresponding to $L_t = (\\tilde{N}_{t+1} - \\tilde{N}_{t})^2$ with $\\lambda = 1$ are provided in Figures \\ref{fig:pois-N=100}, \\ref{fig:pois-N=1000}, \\ref{fig:pois-N=10000} and \\ref{fig:pois-N=100000}. The used sample sizes were $N=100, N=1000, N=10000$ and $N= 100000$. The sample sizes $N=100$ and $N=1000$ resulted complex valued estimates in $45.1\\%$ and $3.0\\%$ of the simulations respectively, whereas with the larger sample sizes all the estimates were real.\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{pois-N=100-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:pois-N=100-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{pois-N=100-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:pois-N=100-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{pois-N=100-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:pois-N=100-l1}\n \\end{subfigure}\n\\caption{Compensated Poisson liquidity with $\\lambda=1$ and $N=100$.}\n\\label{fig:pois-N=100}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{pois-N=1000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:pois-N=1000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{pois-N=1000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:pois-N=1000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{pois-N=1000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:pois-N=1000-l1}\n \\end{subfigure}\n\\caption{Compensated Poisson liquidity with $\\lambda=1$ and $N=1000$.}\n\\label{fig:pois-N=1000}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{pois-N=10000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:pois-N=10000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{pois-N=10000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:pois-N=10000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{pois-N=10000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:pois-N=10000-l1}\n \\end{subfigure}\n\\caption{Compensated Poisson liquidity with $\\lambda=1$ and $N=10000$.}\n\\label{fig:pois-N=10000}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n  \\begin{subfigure}[t]{0.32\\textwidth}\n   \\includegraphics[width=\\textwidth]{pois-N=100000-alpha0.pdf}\n    \\caption{Estimates of $\\alpha_0$.}\n    \\label{fig:pois-N=100000-alpha0}\n  \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{pois-N=100000-alpha1.pdf}\n    \\caption{Estimates of $\\alpha_1$.}\n   \\label{fig:pois-N=100000-alpha1}\n \\end{subfigure}\n  %\n \\begin{subfigure}[t]{0.32\\textwidth}\n    \\includegraphics[width=\\textwidth]{pois-N=100000-l1.pdf}\n    \\caption{Estimates of $l_1$.}\n   \\label{fig:pois-N=100000-l1}\n \\end{subfigure}\n\\caption{Compensated Poisson liquidity with $\\lambda=1$ and $N=100000$.}\n\\label{fig:pois-N=100000}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=11cm]{elo31_TeffMv_meas.ps}\n\\caption{Distribution of the ELODIE.3.1 library\nin the HR diagram, with three metallicity bins differentiated with symbols \nof different colors\n}\n\\label{TeffMv}\n\\end{figure*}\n\nThe ELODIE library is based on spectra retrieved from the Observatoire\nde Haute-Provence archive \\footnote{http:\/\/atlas.obs-hp.fr\/elodie}\n(Moultaka et al. 2004). The spectra were obtained with the ELODIE\nspectrograph attached to the 1.93m telescope, they have a spectral\nresolution of about R=42000 and consists of 67 echelle orders covering\nin total the wavelength range 389.2 to 680 nm without any gap.\n\nSome of the spectra were acquired for the purpose of building a stellar \nlibrary, and others were selected from the archive to complete\nthe coverage in effective temperature (T$_{eff}$), surface gravity\n($log g$) and metallicity ([Fe\/H]) and to achieve the flux calibration\n(by providing enough external comparisons and repeated observations\nof the same stars). \n\nAn early library, not calibrated in flux, was published in Soubiran et al.\n(1998). The first flux-calibrated release of the library was published\nby Prugniel \\& Soubiran (2001), it contained 906 spectra.\nThe last previous release, ELODIE.3 was announced in Prugniel \\& Soubiran\n(2004) and represented a major progress in the coverage of the space of\natmospheric parameters (T$_{eff}$, $log g$ and [Fe\/H]).\n\nThe present version, ELODIE.3.1 is based on the same collection of\nspectra than ELODIE.3.\nFig. 1, 2 and 3 show the distribution of the 1388 stars of the library in\nthe HR diagram, T$_{eff}$ - [Fe\/H] and  T$_{eff}$ - $log g$ planes. \nThe main interest of this new version is the extended wavelength coverage,\nnow including the H \\& K lines, and limited by the blue limit of\nthe spectrograph. Some important improvements are also summarized in the\nnext section.\n\nThe library has been used for population synthesis using the Pegase.HR program\n(Le Borgne et al. 2004), and these models were used to study the stellar\npopulations of galaxies and star clusters. The advantage of these models is\nthe high spectral resolution provided by the library which allows to\nconstrain simultaneously the internal kinematics and the parameters\nof the stellar population (for instance age and metallicity). Thanks\nto this library, the analysis is not anymore limited to the low resolution\nof the spectrophotometric indices. \nSpecific methods for the simultaneous analysis of the kinematics and of the \nstellar population were developed in Ocvirk et al. (2006) and Chilingarian \n(2007). It was found that these approaches of full spectrum fitting\nare increasing the precision on age and metallicity by a factor 3\n(a factor 10 on observing time to get the same precision) (Koleva et al. 2006).\n\nThe comparison between models based on the current release of the library\nand other models are presented in Koleva et al (2007a) and validation\nusing Galactic clusters in Koleva et al. (2007b).\n\nWe are now preparing a further version, ELODIE.4, which will contain about\n5000 spectra. This will allow a better investigation of the effect of\nthe $\\alpha$-element abundances in stellar populations, possibly by\ncoupling the library with a basis of theoretical spectra as presented\nin Prugniel et al. (2007).\n\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=11cm]{elo31_TeffFeH_meas.ps}\n\\caption{Distribution in the T$_{eff}$ - [Fe\/H] plane.\nDwarfs with red dots and giants in green.\n}\n\\label{TeffFeH}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=11cm]{elo31_TeffG_meas.ps}\n\\caption{Distribution in the T$_{eff}$ - $log g$ plane.\nThe color of the symbols distinguish the different metallicity classes.\n}\n\\label{Tefflogg}\n\\end{figure*}\n\n\n\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=11cm]{fig_lick_worthey.ps}\n\\caption{ \nComparison between Lick indices measured on ELODIE library and the \nWorthey et al. (1994) values.\n}\n\\label{lickworthey}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=11cm]{fig_lick_jones.ps}\n\\caption{ \nComparison between Lick indices measured on ELODIE library and the Jones \nlibrary values (Worthey \\& Ottaviani, 1997)\n}\n\\label{lickjones}\n\\end{figure*}\n\n\\section{ Reduction procedure}\nSince the first version of the library, the reduction procedure described \nin Prugniel \\& Soubiran (2001) has been improved in several aspects but the \ngeneral philosophy remains the same. \nThe first steps of the data processing up to the extraction of the orders\nare made with the standard ELODIE pipeline (Baranne et al., 1996).\nThe reconnection of the orders, and the flux calibration are explained\nin  Prugniel \\& Soubiran (2001).\n\nThe specific points modified in the current release are:\n\n\\begin{description}\n\n\\item[Extension of the wavelength range.]\nIn the previous releases the first 4 orders in the blue were not processed\nbecause of the difficulty of their reconnection and calibration due\nto the low signal-to-noise ratio (S\/N). The sensitivity of the detector is\nindeed dramatically dropping near the blue limit.\nWe improved our procedure and we could include the blue region, extending\nto the H \\& K lines.\n\n\\item[Subtraction of the diffuse light.]\nWhen working on the blue region of the spectra, we discovered that the diffuse \nlight on the standard pipeline was often under-subtracted. The consequence\nwas that some sharp lines in the blue were not deep enough.\nThe correction that we applied was validated by comparing with spectra\nfrom other archives or libraries.\n\n\\item[Improvement of the atmospheric parameters.]\nTo use the library for population synthesis, a key factor is to obtain the\natmospheric parameters of the stars with the highest accuracy. In the present\ncase, the atmospheric parameters are first compiled from the literature, and\nan inter-comparison in the library allow to detect inconsistencies and\nto determine 'internal' values: Each wavelength point of the spectra is\nmodeled as polynomials in T$_{eff}$, $log g$ and [Fe\/H] fitted to\nthe compiled parameters and the inversion of this function returns internal\ndeterminations of the parameters (see Prugniel \\& Soubiran 2001, Le Borgne \net al. 2004).\\\\\nOn one side we have updated the compiled input parameters, using recent\npublications and fixing errors, and on the other side we improved the\npolynomial functions used to model the spectra. The present version is\nbetter, but, as seen in Fig.~2 and 3, some artifacts persist in the\ntemperature range 6500-7500K at low metallicity (some input metallicities \nare underestimated).\\\\\nBy comparing the observed spectra with the polynomial model, we get an idea of\nthe quality of the model and of the consistency of the atmospheric parameters.\nThe rms residuals from the 'absolute' model (ie. using the input atmospheric\nparameters) is 2.0\\%. The rms residuals from the 'internal' model (iterated\nusing the atmospheric parameters returned after an inversion with the \n'absolute' model) is 1.4\\%. The decrease of the rms residuals after the \niteration is due to an improvement of the consistency of the atmospheric\nparameters. The final residuals include the effect of the noise and of all\nthe neglected parameters, like rotation, detailed abundances... The\ninternal determination of the atmospheric parameters may be biased with\nrespect to the input values: We checked these biases by averaging them\nin some regions of the parameters space, and did not find evidence for\nclear bias.\n\n\\end{description}\n\nTables 1 \\& 2, and Fig. 4 \\& 5 compare the Lick indices measured on the\nELODIE spectra with those of the Lick (Worthey et al. 1994) and Jones\n(Worthey \\& Ottaviani, 1997) libraries. The comparison with the Lick data \nshows an important spread\ndue to the lower quality of the Lick data, but the comparison with\nJones library reveals an excellent consistency. The zero-points are\nextremely small, and the slope between the two series is close to 1.\nThis test of the slope is critical since it may unveil some data-reduction\nerrors in the cosmic ray clipping, the rebinning or the scattered light\nsubtraction: These three aspects potentially affecting more the \nstrong lines, hence high metallicity stars.\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ l  c c  c}\n\\hline\n\\hline\n    \\multirow{2}*{Index}&   \\multicolumn{3}{c}{Worthey}    \\\\\n\\cline{2-4}\n                        &   offset&    slope& $\\sigma$     \\\\\n\\hline\n    Ca4227              &  0.042  &  0.778  &  0.152       \\\\\n     G4300              &  0.019  &  0.930  &  0.321       \\\\\n    Fe4531              &  0.007  &  0.853  &  0.247       \\\\\n    Fe4668              &  0.023  &  0.935  &  0.470       \\\\\n     Hbeta              &  0.022  &  0.951  &  0.143       \\\\\n    Fe5015              & -0.142  &  0.889  &  0.359       \\\\\n       Mg1              & -0.002  &  0.862  &  0.011       \\\\\n       Mg2              &  0.001  &  0.916  &  0.009       \\\\\n       Mgb              & -0.032  &  0.885  &  0.204       \\\\\n    Fe5270              & -0.036  &  0.980  &  0.253       \\\\\n    Fe5335              & -0.019  &  0.975  &  0.215       \\\\\n    Fe5406              & -0.007  &  0.936  &  0.203       \\\\\n    Fe5782              & -0.012  &  0.733  &  0.119       \\\\\n       NaD              &  0.069  &  0.894  &  0.305       \\\\\n      TiO1              & -0.002  &  0.886  &  0.010       \\\\\n      TiO2              &  0.001  &  0.909  &  0.007       \\\\\n\\hline\n\\end{tabular}\n\\caption{\nComparison with Lick indices from the Lick library. Col. 1: Designation of the Lick index; Col. 2, 3: Index(Elodie) = offset + slope $\\times$ Index(Worthey);\nCol. 4: rms dispersion from the linear fit.}\n\\end{table}\n\n\\section{Data access}\nThe spectra are available as fits files. Three gzipped tarfiles, corresponding \nto \nthe R=10000, R=42000 libraries can be downloaded, as well as \nthe table of measurements\ncontaining the stellar parameters estimated internally and the measured\nLick indices.  \nThe description of the files and the specific keywords are described\non the ELODIE.3.1 web page.\n\nTwo versions of the library are constructed. The high resolution (R $\\approx$\n42000) corresponds to the nominal resolution of the spectrograph, and the\nlow resolution (FWHM = 0.55 \\AA, R $\\approx$ 10000) has been produced\nby a gaussian broadening. The high resolution version is provided\nnormalized to the pseudo-continuum while the low resolution is in \nphysical flux with an absolute calibration (physical units) scaled\nusing Tycho photometry. Accessing the library using the HyperLeda \ndatabase\\footnote{http:\/\/leda.univ-lyon1.fr}\nallows conversions of the flux normalization, or various convolutions\nand rebinnings.\n\nUsers of this library are kindly requested to cite in their publications\nthe original A\\&A publication (Prugniel \\& Soubiran 2001) as well as the \npresent astro-ph announcement. The proper denomination of the current\nversion is: ELODIE.3.1.\n\nWe are encouraging reports on the usages of this library and encountered\ndifficulties. They will help us to continue to improve the quality.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ l  c c  c}\n\\hline\n\\hline\n    \\multirow{2}*{Index}&     \\multicolumn{3}{c}{Jones}    \\\\\n\\cline{2-4}\n                        &   offset&    slope& $\\sigma$     \\\\\n\\hline\n    Ca4227              &  0.025  &  0.902  &  0.075       \\\\\n     G4300              &  0.079  &  0.962  &  0.187       \\\\\n     Hbeta              &  0.003  &  0.967  &  0.072       \\\\\n    Fe5015              &  0.006  &  0.967  &  0.166       \\\\\n       Mg1              & -0.003  &  0.978  &  0.011       \\\\\n       Mg2              &  0.000  &  0.961  &  0.007       \\\\\n       Mgb              &  0.021  &  0.997  &  0.072       \\\\\n    Fe5270              &  0.012  &  0.992  &  0.057       \\\\\n    Fe5335              &  0.001  &  0.952  &  0.061       \\\\\n    Fe5406              &  0.008  &  0.998  &  0.047       \\\\\n\\hline\n\\end{tabular}\n\\caption{\nComparison with Lick indices from the Jones library. Col. 1: Designation of the Lick index; Col. 2, 3: Index(Elodie) = offset + slope $\\times$ Index(Jones);\nCol. 4: rms dispersion from the linear fit.\n}\n\\end{table}\n\n\n\\noindent {\\bf Acknowledgments.}\nWe thank Observatoire de Haute-Provence, and in particular Sergio Ilovaisky, \nfor maintening a public archive of their observations.\nMany improvements of this library were possible thanks to the feed-back\nof the users.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}