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We investigate hermitian Yang–Mills connections for pullback vec- +tor bundles on blow-ups of K¨ahler manifolds along submanifolds. Under some +mild asumptions on the graded object of a simple and semi-stable vector bun- +dle, we provide a necessary and sufficent numerical criterion for the pullback +bundle to admit a sequence of hermitian Yang–Mills connections for polarisa- +tions that make the exceptional divisor sufficiently small, and show that those +connections converge to the pulled back hermitian Yang-Mills connection of +the graded object. +1. Introduction +A corner stone in gauge theory is the Hitchin–Kobayashi correspondence ([17, +20, 30, 12]). This celebrated generalisation of the Narasimhan and Seshadri the- +orem asserts that a holomorphic vector bundle over a K¨ahler manifold carries an +Hermite–Einstein metric if and only if it is polystable in the sense of Mumford and +Takemoto ([22, 29]). The interplay between the differential geometric side, her- +mitian Yang–Mills connections (HYM for short) that originated from physics, and +the algebro-geometric side, the stability notion motivated by moduli constructions, +has had many applications and became a very fertile source of inspiration. Given +that HYM connections are canonically attached to polystable vector bundles, it is +natural to investigate their relations to natural maps between vector bundles, such +as pullbacks. In this paper, we address the problem of pulling back HYM connec- +tions along blow-ups. While the similar problem for extremal K¨ahler metrics has +seen many developments in the past ten years [1, 2, 3, 28, 26, 8], relatively little +seems to be known about the behaviour of HYM connections under blow-ups [6, 9]. +In this paper, under some mild asumptions, we solve the problem for pullback of +semi-stable vector bundles on blow-ups along smooth centers. +Let π : X′ → X be the blow-up of a polarised K¨ahler manifold (X, [ω]) along a +submanifold Z ⊂ X, and E′ = π∗E the pullback of a holomorphic vector bundle +E → X. For 0 < ε ≪ 1, Lε := π∗[ω] − εc1(Z′) defines a polarisation on X′, where +we set Z′ = π−1(Z) the exceptional divisor. There are obstructions for E′ to admit +HYM connections with respect to ωε ∈ c1(Lε), with 0 < ε ≪ 1. In particular, E +should be simple and semi-stable with respect to [ω] (see Section 2.3). In the latter +case, E admits a Jordan–Holder filtration by semi-stable sheaves with polystable +graded object Gr(E) (see Section 2.2 for definitions). A further obstruction comes +then from subsheaves of E arising from Gr(E). While those sheaves have the same +slope as E, their pullbacks to X′ could destabilise E′. Our main result asserts that +those are actually the only obstructions for E′ to carry a HYM connection, under +some mild asumptions on Gr(E). +2010 Mathematics Subject Classification. Primary: 53C07, Secondary: 53C55, 14J60. +1 + +2 +A. CLARKE AND C. TIPLER +Recall that a semi-stable holomorphic vector bundle E → (X, [ω]) is said to be +sufficiently smooth if its graded object Gr(E) is locally free. Let E[ω] denote the set +of all subbundles of E arising in a Jordan–Holder filtration for E, or equivalently of +same slope as E with respect to [ω]. For F ∈ E[ω], denote by µLε(F) = c1(π∗F)·Ln−1 +ε +rank(F) +the slope of π∗F on (X′, Lε). +Theorem 1.1. Let E → X be a simple sufficiently smooth semi-stable holomorphic +vector bundle on (X, [ω]). Assume that the stable components of Gr(E) are pairwise +non-isomorphic. Then, there exists ε0 > 0 and a sequence of HYM connections +(Aε)ε∈(0,ε0) on π∗E with respect to (ωε)ε∈(0,ε0) if and only if +(1.1) +∀ F ∈ E[ω], µLε(F) < +ε→0 µLε(E). +In that case, if A denotes a HYM connection on Gr(E) with respect to ω, then +(Aε)ε∈(0,ε0) can be chosen so that Aε −→ +ε→0 π∗A in any Sobolev norm. +In the statement, the expression µLε(F) < +ε→0 µLε(E) means that the first non- +zero term in the ε-expansion for µLε(E) − µLε(F) is strictly positive. +Remark 1.2. Simplicity, semi-stability and condition (1.1) are necessary to pro- +duce the connections (Aε) from Theorem 1.1. The other two asumptions on Gr(E) +are technical. Assuming Gr(E) to be locally free enables to see E as a smooth com- +plex deformation of Gr(E) and to work with the various connections on the same +underlying complex vector bundle. We should warn the reader though that if one +drops this asumption, Condition (1.1) might not be enough to ensure semi-stability +of π∗E on (X′, Lε) (see the extra conditions in [23, Theorem 1.10]). On the other +hand, the asumption on Gr(E) having no pairwise isomorphic components is purely +technical, and ensures that its automorphism group, that will provide obstructions +in the perturbative theory, is abelian. +We now list some corollaries of Theorem 1.1. First, the stable case : +Corollary 1.3. Let E → X be a stable holomorphic vector bundle on (X, [ω]) and +let A be a HYM connection on E. Then, there exists ε0 > 0 and a sequence of HYM +connections (Aε)ε∈(0,ε0) on π∗E with respect to (ωε)ε∈(0,ε0) such that Aε → +ε→0 π∗A +in any Sobolev norm. +For the semi-stable case, Condition (1.1) reduces to a finite number of intersec- +tion product computations. One interesting feature comes from the second term in +the expansion of µLε(E). It is the opposite of the slope of the restriction of E to +Z. The following formula is proved in [23, Section 4.1], where m = dim(Z) : +(1.2) +µLε(E) = µL(E) − +�n − 1 +m − 1 +� +µL|Z(E|Z)εn−m + O(εn−m+1). +We then have : +Corollary 1.4. Let E → X be a simple sufficiently smooth semi-stable holomorphic +vector bundle on (X, [ω]). Assume that the stable components of Gr(E) are pairwise +non-isomorphic. Denote by A an HYM connection on E. If +(1.3) +∀ F ∈ E[ω], µL|Z(E|Z) < µL|Z(F|Z), +then, there exists ε0 > 0 and a sequence of HYM connections (Aε)ε∈(0,ε0) on π∗E +with respect to (ωε)ε∈(0,ε0) converging to π∗A in any Sobolev norm. + +BLOWING-UP HYM CONNECTIONS +3 +Condition (1.3) was checked on explicit examples in [23, Section 4.5] to produce +stable perturbations of tangent sheaves by blow-ups, and our result provides infor- +mation on the associated connections and their asymptotic behaviour. Note that +by Mehta–Ramanathan theorem [21], if [ω] = c1(L) is integral, and if Z is a generic +intersection of divisors in linear systems |Lk|, then E|Z is semi-stable as soon as E +is. In that case, Condition (1.3) cannot be satisfied, and it seems unlikely that Con- +dition (1.1) will hold true. Hence, blowing-up such subvarieties tend to destabilise +a semi-stable bundle. +In general, we expect that it should not be too hard to obtain stability of suffi- +ciently smooth pulled back bundles under condition (1.1) with purely algebraic +methods. +However, we emphasize that the Hitchin–Kobayashi correspondence +doesn’t provide any information on the asymptotic behaviour of the associated +HYM connections, which is then the main content of Theorem 1.1. Nevertheless, we +state the following corollary, that extends [23, Theorem 1.10] to a non-equivariant +situation: +Corollary 1.5. Let E → X be a simple sufficiently smooth semi-stable holomorphic +vector bundle on (X, [ω]). Assume that the stable components of Gr(E) are pairwise +non-isomorphic. Then, there exists ε0 > 0 such that π∗E → (X′, Lε) is +(i) stable if and only if for all F ∈ E[ω], µLε(F) < +ε→0 µLε(E), +(ii) semi-stable if and only if for all F ∈ E[ω], µLε(F) ≤ +ε→0 µLε(E), +(iii) unstable otherwise. +Finally, we comment on previous related works. Theorem 1.1 extends results +from [6, 9] where blow-ups of HYM connections along points are considered. In the +present paper, we consider blow-ups along any smooth subvariety, and also cover +the semi-stable situation, which is technically more involved due to the presence of +automorphisms of the graded object that obstruct the linear theory. While [9] is a +gluing construction as in the similar problem of producing extremal K¨ahler metrics +on blow-ups [2, 3, 28, 26, 8], one of the key feature in our approach is to apply +directly the implicit function theorem to reduce to (an ε dependent family of) finite +dimensional GIT problems on a Kuranishi space parametrising small deformations +of Gr(E), as in [27, 8]. We then use the new technology developed in [24] to control +the perturbations of the associated moment maps when ωε varries. This is where +our hypothesis on Aut(Gr(E)) being abelian is used. +The main new technical +input comes from the fact that the underlying smooth manifold X is fixed in [24], +while it varries with the blow-up, which requires a carefull analysis of the operator +introduced to apply the implicit function theorem. +Outline: In Section 2, we recall basic material about HYM connections and stabil- +ity. We then perform in Section 2.3 the analysis of the linear theory on the blow-up. +Relying on this, in Section 3 we explain how to reduce the problem to finding zeros +of finite dimensional moment maps. Then, we conclude the proof of Theorem 1.1 +and its corollaries in Section 4. +Acknowledgments: The authors benefited from visits to LMBA and Gotheborg +University; they would like to thank these welcoming institutions for providing +stimulating work environments. The idea of this project emerged from discussions +with Lars Martin Sektnan, whom we thank for sharing his ideas and insight. CT + +4 +A. CLARKE AND C. TIPLER +is partially supported by the grants MARGE ANR-21-CE40-0011 and BRIDGES +ANR–FAPESP ANR-21-CE40-0017. +2. Preliminaries +In Sections 2.1 and 2.2 we introduce the notions of HYM connections and slope +stability, together with some general results, and refer the reader to [18] and [16]. +From Section 2.3 we start to specialise the discussion to blow-ups. In particular, +in Section 2.3.2, we provide various asymptotic expressions for the linearisation of +the HYM equation on the blow-up. Those results will be used in Section 3. +2.1. The hermitian Yang–Mills equation. Let E → X be a holomorphic vector +bundle over a compact K¨ahler manifold X. A hermitian metric on E is Hermite– +Einstein with respect to a K¨ahler metric with K¨ahler form ω if the curvature +Fh ∈ Ω2 (X, End E) of the corresponding Chern connection satisfies +Λω (iFh) = c IdE +(2.1) +for some real constant c. Equivalently, if h is some hermitian metric on the smooth +complex vector bundle underlying E, a hermitian connection A on (E, h) is said to +be hermitian Yang–Mills if it satisfies +� +F 0,2 +A += +0, +Λω (iFA) += +c IdE . +The first equation of this system implies that the (0, 1)-part of A determines a +holomorphic structure on E, while the second that h is Hermite–Einstein for this +holomorphic structure. We will try to find hermitian Yang–Mills connections within +the complex gauge group orbit, which we now define. The (hermitian) complex +gauge group is +G C(E, h) = Γ (GL (E, C)) ∩ Γ (EndH(E, h)) , +where EndH(E, h) stands for the hermitian endomorphisms of (E, h). Note that if +¯∂ is the Dolbeault operator defining the holomorphic structure on E, then f ◦ ¯∂◦f −1 +defines a biholomorphic complex structure on E. Let dA = ∂A + ¯∂A be the Chern +connection of (E, h) with respect to the original complex structure (that is ¯∂A = ¯∂). +Then the Chern connection Af of h with respect to f ◦ ¯∂ ◦ f −1 is +dAf = (f ∗)−1 ◦ ∂A ◦ (f ∗) + f ◦ ¯∂ ◦ f −1. +Solving the hermitian Yang–Mills equation is equivalent to solving +Ψ(s) = c IdE +where +Ψ : +Lie(G C(E, h)) +−→ +Lie(G C(E, h)) +s +�−→ +iΛω(FAexp(s)), +and where Lie(G C(E, h)) := iΓ(EndH(E, h)) is the tangent space to G C(E, h) at +the identity. For a connection A on E, the Laplace operator ∆A is +∆A = iΛω +�¯∂A∂A − ∂A ¯∂A +� +. +(2.2) +If AEnd E denote the connection induced by A on End E, then : + +BLOWING-UP HYM CONNECTIONS +5 +Lemma 2.1. If A is the Chern connection of (E, ∂, h), the differential of Ψ at +identity is +dΨIdE = ∆AEnd E. +If moreover A is assumed to be hermitian Yang–Mills, then the kernel of ∆AEnd E +acting on Γ(End(E)) is given by the Lie algebra aut(E) of the space of automor- +phisms Aut(E) of (E, ∂). +The last statement about the kernel follows from the K¨ahler identities and the +Akizuki-Nakano identity that imply ∆AEnd E = ∂∗ +A∂A + ¯∂∗ +A ¯∂A, the two terms of +which are equal if A is Hermitian Yang-Mills. The operator ∆AEnd E being elliptic +and self-adjoint, aut(E) will then appear as a cokernel in the linear theory for +perturbations of hermitian Yang–Mills connections. +2.2. Slope stability. We recall some basic facts about slope stability, as intro- +duced by [22, 29], and refer the interested reader to [16] for a detailed treatment. +We denote here L := [ω] the polarisation of the n-dimensional K¨ahler manifold X. +Definition 2.2. For E a torsion-free coherent sheaf on X, the slope µL(E) ∈ Q +(with respect to L) is given by the intersection formula +(2.3) +µL(E) = degL(E) +rank(E) , +where rank(E) denotes the rank of E while degL(E) = c1(E) · Ln−1 stands for its +degree. Then, E is said to be slope semi-stable (resp. slope stable) with respect to +L if for any coherent subsheaf F of E with 0 < rank(F) < rank(E), one has +µL(F) ≤ µL(E) ( resp. µL(F) < µL(E)). +A direct sum of slope stable sheaves of the same slope is said to be slope polystable. +In this paper, we will often omit “slope” and simply refer to stability of a sheaf, +the polarisation being implicit. We will make the standard identification of a holo- +morphic vector bundle E with its sheaf of sections, and thus talk about slope sta- +bility notions for vector bundles as well. In that case slope stability relates nicely +to differential geometry via the Hitchin–Kobayashi correspondence : +Theorem 2.3 ([17, 20, 30, 12]). There exists a Hermite–Einstein metric on E with +respect to ω if and only if E is polystable with respect to L +We will be mostly interested in semi-stable vector bundles. A Jordan–H¨older +filtration for a torsion-free sheaf E is a filtration by coherent subsheaves: +0 = F0 ⊂ F1 ⊂ . . . ⊂ Fℓ = E, +(2.4) +such that the corresponding quotients, +Gi = +Fi +Fi−1 +, +(2.5) +for i = 1, . . . , ℓ, are stable with slope µL(Gi) = µL(E). In particular, the graded +object of this filtration +(2.6) +Gr(E) := +l +� +i=1 +Gi +is polystable. From [16, Section 1], we have the standard existence and uniqueness +result: + +6 +A. CLARKE AND C. TIPLER +Proposition 2.4. Any semi-stable coherent torsion-free sheaf E on (X, L) admits +a Jordan–H¨older filtration, and the graded object Gr(E) of such filtrations is unique +up to isomorphism. +When E is locally-free and semi-stable, we say that it is sufficiently smooth if +Gr(E) is locally-free. In that case, we denote E[ω] the set of holomorphic subbundles +of E built out of successive extensions of some of the stable components of Gr(E). +Equivalently, E[ω] is the set of holomorphic subbundles of E arising in a Jordan- +Holder filtration for E. Finally, we recall that a necessary condition for E to be +stable is simplicity, that is Aut(E) = C∗ · IdE. +2.3. Geometry of the blow-up. We consider now Z ⊂ X a m-dimensional com- +plex submanifold of codimension r = n − m ≥ 2 and the blow-up map +π : BlZ(X) → X. +We will denote by X′ = BlZ(X) the blown-up manifold and by Z′ = π−1(Z) the +exceptional divisor. We denote by +Lε := π∗L − ε[Z′] +a polarisation on X′, for 0 < ε ≪ 1. Let E → X be a holomorphic vector bundle, +and denote by E′ = π∗E the pulled back bundle. For any holomorphic subbundle +F ⊂ E, the intersection numbers µLε(π∗E)−µLε(π∗F) admit expansions in ε, with +first term given by µL(E) − µL(F). For that reason, given the Hitchin–Kobayashi +correspondence in Theorem 2.3, semi-stability of E on (X, L) is a necessary con- +dition for its pullback E′ to admit an HYM connection with respect to a K¨ahler +metric in Lε, for all 0 < ε ≪ 1. Another necessary condition is simplicity of E′, +which, by Hartogs’ theorem, is equivalent to simplicity of E. Then, natural can- +didates to test for stability of E′ are given by the pullbacks of elements in E[ω], +and Condition (1.1) clearly is necessary for E′ to be stable in the polarisations +we consider, and thus to admit an HYM connection. Hence, we will assume E to +be simple, semi-stable, and to satisfy (1.1). We now turn back to the differential +geometry of the blow-up. +2.3.1. Decomposition on spaces of sections. We have a commutative diagramm: +Z′ +ι +−→ +X′ +↓ +↓ +Z +ι0 +−→ +X +where ι0 and ι denote the inclusions, while the vertical arrows are given by the +projection map π. We then have a pullback map on sections +π∗ : Γ(X, End(E)) −→ Γ(X′, End(π∗E)) +as well as a restriction map : +ι∗ : Γ(X′, End(π∗E)) −→ Γ(Z′, End(ι∗π∗E)). +Our goal now is to fit those maps in a short exact sequence, that will in the end split +the space Γ(X′, End(π∗E)). If NZ = T X|Z/T Z denotes the normal bundle of Z in +X, then Z′ ≃ P(NZ), and we can fix a (1, 1)-form λ ∈ c1(OP(NZ)(1)) that restricts +to K¨ahler metrics on the fibers of P(NZ) → Z. We also fix a K¨ahler form ω ∈ c1(L) +on X, and consider its restriction to Z. We then have a K¨ahler CPr−1-fibration : +π : (Z′, λ) −→ (Z, ω). + +BLOWING-UP HYM CONNECTIONS +7 +By averaging along fibers as described in [25, Section 2.3], we obtain a splitting +(2.7) +Γ(Z′, End(ι∗π∗E)) = π∗(Γ(Z, End(ι∗ +0E))) ⊕ Γ0(Z′, End(ι∗π∗E)). +We will omit the ι∗ and π∗ to simplify notation. Using the projection on the second +factor +p0 : Γ(Z′, End(E)) → Γ0(Z′, End(E)) +in (2.7), we deduce a short exact sequence : +0 −→ Γ(X, End(E)) +π∗ +−→ Γ(X′, End(E)) +p0◦ι∗ +−→ Γ0(Z′, End(E)) −→ 0. +We can actually split this sequence by mean of a linear extension operator +ι∗ : Γ0(Z′, End(E)) −→ Γ(X′, End(E)) +such that +p0 ◦ ι∗ ◦ ι∗ = Id. +This can be done using bump functions and a standard partition of unity argument. +The outcome is an isomorphism : +(2.8) +Γ(X′, End(E)) +−→ +Γ(X, End(E)) ⊕ Γ0(Z′, End(E)) +s +�−→ +(s − ι∗ ◦ p0 ◦ ι∗s , p0 ◦ ι∗s), +with inverse map (sX, sZ) �→ (π∗sX + ι∗sZ). This splits the Lie algebra of gauge +transformations, and will be used to identify contributions coming from X and from +Z′ in the ε-expansion of the linearisation, which we describe in the next section. +From now on, by abuse of notations, we will consider the spaces Γ(X, End(E)) +and Γ0(Z′, End(E)) as subspaces of Γ(X′, End(π∗E)), and denote s = sX + sZ the +decomposition of an element s ∈ Γ(X′, End(E)). +2.3.2. Decomposition of the Laplace operator. We extend λ to a closed (1, 1)-form +over X′ as in [31, Section 3.3] and consider the family of K¨ahler metrics on X′: +ωε = π∗ω + ελ ∈ c1(Lε), 0 < ε ≪ 1. +Let A be a Hermitian connection on E, which we pull back to X′ and extend to +the bundle End(π∗E). We will now study the Laplace operator +∆εs = iΛε(¯∂A∂A − ∂A ¯∂A)s +acting on the various components of s = sX + sZ ∈ Γ(X′, End(E)), where Λε is +the Lefschetz operator for the metric ωε. For this, we need to introduce an elliptic +operator on Z′. The vertical Laplace operator, denoted +∆V : Γ0 (Z′, End(E)) → Γ0 (Z′, End(E)) , +is the operator defined by the following procedure. Let σ ∈ Γ0(Z′, End(E)). Over a +point x ∈ Z, take the restriction σx of σ to Z′ +x = π−1(x), and consider σx as a map +to Cp with components σi +x in a trivialisation π∗ End(E)x ∼= Cp of the restriction of +π∗ End(E) to the fibre Z′ +x of Z′ → Z. Define +(∆V (σ))i +x = ∆(λ)|Z′x +� +σi +x +� +, +for ∆λ the Laplacian of the K¨ahler form λ on Z′ +x. Then glue together to form +a section of π∗ End(E). As in [25, Section 4.1], one easily obtains that this con- +struction is independent on the trivialisation chosen, and sends smooth sections to +smooth sections. In the following Lemma, the supscript l (or l + 2) stands for the +Sobolev completion with respect to some L2,l Sobolev norm, where those norms + +8 +A. CLARKE AND C. TIPLER +can be produced out of the metrics ω, λ and any metric h on E, together with the +covariant derivatives given by A. +Lemma 2.5. [25, Section 4.1] The vertical Laplacian +∆V : Γ0 (Z′, End(E))l+2 → Γ0 (Z′, End(E))l +is invertible. +In the following statements, if A denotes a second order operator acting on +sections, then in an expression of the form +A(σ) = σ0 + εσ1 + . . . + εd−1σd−1 + O(εd) +the term O(εd) will stand for σd ·εd, where σd is a section whose L2,l Sobolev norm +is bounded by the L2,l+2 Sobolev norm of σ. +Lemma 2.6. If sZ = ι∗σZ for σZ ∈ Γ(Z′, End(E)), then +(p0 ◦ ι∗)∆ε(ι∗σZ) = ε−1∆VσZ + O(1). +Proof. We introduce the operator D given by +DsZ = i(¯∂A∂A − ∂A ¯∂A)sZ. +The Laplacian ∆ε satisfies on X′ : +∆εsZ ωn +ε = nDsZ ∧ ωn−1 +ε +, +or equivalently +∆εsZ = n DsZ ∧ (ω + ελ)n−1 +(ω + ελ)n +. +We note that ω is a K¨ahler form on X, but on X′ is degenerate along the fibre +directions of the submanifold Z′. +Then (i∗ω)m+1 = 0 ∈ Ω2(m+1)(Z′), and at +x ∈ Z′ ⊆ X′, ωm+2 = 0. Then, expanding (ω + ελ)n−1 and (ω + ελ)n gives +ι∗∆εsZ = (n − m − 1)ε−1 DsZ ∧ ωm+1 ∧ λn−m−2 +ωm+1 ∧ λn−m−1 ++ O(1). +Restricting to Z′, the connection 1-forms of A vanish, so ι∗DsZ = i∂ ¯∂σZ, acting +on the coefficient functions of σZ. On the other hand, by considering a convenient +orthonormal frame at x ∈ Z′, we see that ι∗∆ει∗σZ = ε−1∆VσZ + O(1). +□ +In the next lemma, we denote ∆εsZ = (∆εsZ)X + (∆εsZ)Z the decomposition +according to (2.8). +Lemma 2.7. For sZ = ι∗σZ with σZ ∈ Γ(Z′, End(E)), we have +(∆εsZ)X = O(1). +Proof. By definition, (∆εsZ)X = π∗φ for some φ ∈ Γ(X, End(E)). As we also have +(∆εsZ)X += +(Id − ι∗(p0 ◦ ι∗))ΛεDsZ, +we deduce that the section φ is the continuous extension of π∗(Id−ι∗(p0◦ι∗))ΛεDsZ +across Z ⊆ X. On X′ \ Z′ we have +ΛεDsZ += +nDsZ ∧ (ωn−1 + O(ε)) +ωn + O(ε) += O(1). +As π∗(Id − ι∗(p0 ◦ ι∗)) is O(1), the result follows. +□ + +BLOWING-UP HYM CONNECTIONS +9 +From the previous two lemmas, in the decomposition +s = sX + sZ, +∆εsZ also lies in the subspace Γ0(Z′, End(E)) ⊆ Γ(X′, End(E)) to higher order in +ε. For sX ∈ Γ(X, End(E)), +∆εsX = (∆εsX)X + (∆εsX)Z +where (∆εsX)Z = ι∗(p0 ◦ ι∗)∆εsX. We first consider ι∗∆εsX. +Lemma 2.8. For sX = π∗σX ∈ Γ(X, End(E)) ⊆ Γ(X′, End(E)), +ι∗∆εsX = (m + 1)DsX ∧ ωm ∧ λn−m−1 +ωm+1 ∧ λn−m−1 ++ O(ε). +Proof. Firstly, sX = π∗σX, and the connection A is pulled back from X, so DsX +is basic for the projection to X and Ds ∧ ωm+1 = 0 at points in Z′. Secondly, we +note that ωm+1 ∧λn−m−1 is a volume form on X′, in a neighbourhood of Z′. Then, +the result follows similarly to the previous lemma. +□ +For the final term (∆εsX)X, we introduce ∆X the Laplace operator of A on +End(E) → (X, ω): +∆X : +Γ (X, End(E)) +→ +Γ (X, End(E)) +σ +�→ +iΛω(¯∂A∂A − ∂A ¯∂A)σ. +Lemma 2.9. For sX = π∗σX ∈ Γ(X, End(E)) ⊆ Γ(X′, End(E)), +(∆εsX)X = π∗(∆XσX) + O(ε). +Proof. There is φ ∈ Γ(X, End(E)) such that (∆εsX)X = π∗φ. The element φ can be +identified as the lowest order term in the asymptotic expansion in ε of (∆επ∗σX)X. +However, we have at x ∈ X′ \ Z′ : +∆επ∗σX = nDπ∗σX ∧ (ω + ελ)n−1 +(ω + ελ)n += nπ∗ DσX ∧ ωn−1 +ωn ++ O(ε) +so we see that the lowest order term in the expansion of (∆επ∗σX)X is ∆XσX. +□ +Summarizing the above calculations, with respect to the decomposition s = +sX + sZ produced by (2.8), the operator ∆ε takes the form +(2.9) +� ∆X +0 +L +ε−1∆V +� +plus higher order terms, for some second order operator L. In the next section, we +will apply the previous lemmas and the resulting form for ∆ε to the pullback of a +HYM connection A0 on the graded object Gr(E) of E. +3. The perturbation argument +The goal of this section is to reduce the problem of finding a zero for the operator +s �→ iΛωε(FAexp(s)) − cεId in a gauge group orbit to a finite dimensional problem. +The ideas here go back to [13, 27], and our framework will be that of [5]. + +10 +A. CLARKE AND C. TIPLER +3.1. Kuranishi slice. We start from a simple semi-stable and sufficiently smooth +holomorphic vector bundle E on (X, L), with L = [ω]. Denote by Gr(E) = �ℓ +i=1 Gi +the associated polystable graded object, with stable components Gi. We let ∂0 be +the Dolbeault operator of Gr(E). The automorphism group G := Aut(Gr(E)) is a +reductive Lie group with Lie algebra g := aut(Gr(E)) and compact form K ⊂ G, +with k := Lie(K). The Dolbeault operator ∂E on E is given by +∂E = ∂0 + γ +where γ ∈ Ω0,1(X, Gr(E)∗ ⊗ Gr(E)) can be written +γ = +� +i 0 and a continuously diffen- +rentiable map +ˇΦ : [0, ε0) × B → Ω0,1(X′, End(Gr(E)))l +such that for all (ε, b) ∈ [0, ε0) × B, if ˇAε,b is the Chern connection of (π∗∂0 + +ˇΦ(ε, b), h) : +(1) π∗∂0 + ˇΦ(ε, b) and π∗∂0 + �Φ(b) induce isomorphic holomorphic structures. +(2) ΛεiF ˇ +Aε,b ∈ k. +Remark 3.3. By elliptic regularity, elements in the image of ˇΦ will actually be +smooth. However, regularity of the map ˇΦ is with respect to the L2,l Sobolev norm. +We will use the implicit function theorem to prove Proposition 3.2, and will need +the following lemma, where we still denote A0 its pullback to π∗Gr(E), and use the +notation AsX+εsZ +0 +for Aexp(sX+εsZ) +0 +. +Lemma 3.4. The map : +Ψ : [0, ε0) × Γ(X, EndH(E))l+2 × Γ0(Z′, EndH(E))l+2 +−→ +Ω0(X′, EndH(E))l, +(ε , sX , sZ) +�→ +ΛεFA +sX +εsZ +0 +− cεId +is continuously differentiable. +Above, the topological constants cε are given by +cε = +2πn +volωε(X′) +� +c1(E) ∪ [ωε]n−1� +[X′] +rank(E) +. +Proof. Note first that for ε = 0, Ψ(0, sX, sZ) = π∗(ΛωFA +sX +0 +− c0 IdE) and is well +defined. Then, recall that if f = exp(s) for s ∈ Γ(X′, EndH(E)), the curvature of +f · A0 is given by +FAs +0 = Ff·A0 = FA0 + (¯∂∂ − ∂ ¯∂)s + (∂s − ¯∂s) ∧ (∂s − ¯∂s), + +12 +A. CLARKE AND C. TIPLER +where ∂ and ¯∂ stand for the (1, 0) and (0, 1) components of dA0 (see e.g. [5][Section +1]). In particular, taking s = sX + εsZ, +FAs +0 += +FA0 + (¯∂∂ − ∂ ¯∂)sX + ε(¯∂∂ − ∂ ¯∂)sZ + (∂sX − ¯∂sX) ∧ (∂sX − ¯∂sX) ++ε(∂sX − ¯∂sX) ∧ (∂sZ − ¯∂sZ) + ε(∂sZ − ¯∂sZ) ∧ (∂sX − ¯∂sX) ++ε2(∂sZ − ¯∂sZ) ∧ (∂sZ − ¯∂sZ). +That is, ignoring the first term FA0, there are six remaining terms that we denote +F i +As, for i = 1, . . . 6. For each term we consider the factors coming from Z′ and +from X (using (2.8)) in ΛεF i +As and can conclude that Ψ is smooth. For example, +for the term F 2 +As = ε(¯∂∂ − ∂ ¯∂)sZ, +ΛεF 2 +As += +nεDsZ ∧ (ω + ελ)n−1 +(ω + ελ)n +, +ι∗ΛεF 2 +As += +n +εDsZ ∧ +�� n−1 +m+1 +� +ωm+1 ∧ (ελ)n−m−2 + O(εn−m−1) +� +� +n +m+1 +� +ωm+1 ∧ (ελ)n−m−1 + O(εn−m) +, += +(n − m − 1)DsZ ∧ +� +ωm+1 ∧ λn−m−1 + O(ε) +� +ωm+1 ∧ λn−m−1 + O(ε) +, +noting that here O(ε) denotes a polynomial in ε with coefficients 2n-forms on a +neighbourhood of Z′, such that O(0) = 0. We also note that ωm+1 ∧ λn−m−1 is a +volume form on a neighbourhood of Z′. We conclude that +(ΛεF 2 +As)Z = ι∗(p0 ◦ ι∗)ΛεF 2 +As +is a smooth function of (ε, sZ) with values in Γ0(Z′, End(E)). +The X-component of ΛεF 2 +As, +(ΛεF 2 +As)X += +(Id − ι∗(p0 ◦ ι∗))ΛεF 2 +As, +is of the form π∗φ for some φ ∈ Γ(X, End(E)). +The section φ is given as the +continuous extension of π∗(Id − ι∗(p0 ◦ ι∗))ΛεF 2 +As across Z ⊆ X. On X′ \ Z′ we +have +ΛεF 2 +As += +nDsZ ∧ (ωn−1 + O(ε)) +ωn + O(ε) +, +which depends smoothly on sZ and ε. As π∗(Id − ι∗(p0 ◦ ι∗)) is linear, φ depends +smoothly on these variables too. +Using that sX is a pulled back section, at points in Z′ we have DsX ∧ωm+1 = 0, +from which we deduce ι∗ΛεF 1 +As = O(1) and ι∗ΛεF 3 +As = O(1). This shows, as for +(ΛεF 2 +As)Z, that (ΛεF 1 +As)Z and (ΛεF 3 +As)Z are C1. The other terms F i +As can be dealt +with in a similar manner. +□ +Proof of Proposition 3.2. For b ∈ B, we will denote by Ab the Chern connection +associated to (π∗∂0 + �Φ(b), h), where h = π∗h0. +Note that in particular A0 is +the pullback of a HYM connection on Gr(E). The aim is to apply the implicit +function theorem to perturb Ab along gauge orbits in order to satisfy point (2) of +the statement. The key will be to consider small perturbations along the exceptional +divisor. Recall the splitting from Section 2.3.1 induced by the operator ι∗: +iΓ(X′, EndH(Gr(E), h)) = iΓ(X, EndH(Gr(E), h)) ⊕ iΓ0(Z′, EndH(Gr(E), h)), + +BLOWING-UP HYM CONNECTIONS +13 +that we will simply denote +Γ(X′) = Γ(X) ⊕ Γ0(Z′). +For (sX, sZ) ∈ Γ(X) ⊕ Γ0(Z′), and ε small enough, we define +Ab(ε, sX, sZ) = AsX+εsZ +b +, +where sX + εsZ stands for π∗sX + ε ι∗sZ ∈ Γ(X′). By the regularity of �Φ, the +assignment (b, ε, sX, sZ) �→ Ab(ε, sX, sZ)− A (resp. (b, ε, sX, sZ) �→ FAb(ε,sX,sZ)) is +smooth from B ×[0, ε0)×Γ(X′)l to Ω1(X′, End(E))l−1 (resp. Ω2(X′, End(E))l−2), +for any ε0 small enough. Arguing as in Lemma 3.4, using the fact that the pertur- +bations along Z′ are O(ε), we deduce that the operator +�Ψ : +B × [0, ε0) × Γ(X′)l +→ +Γ(X′)l−2 +(b, ε, sX, sZ) +�→ +ΛεiFAb(ε,sX,sZ) − cε IdE +is a C1 map. As A0 is HYM on Gr(E) → X, we have �Ψ(0) = 0. By the various +lemmas of Section 2.3.2, its differential in the (sX, sZ) direction at zero is given by +the map +Γ(X)l × Γ0(Z′)l +→ +Γ(X)l−2 × Γ0(Z′)l−2 +(sX, sZ) +�→ +� ∆XsX +0 +∗ +∆VsZ +� +which, from Lemma 2.1 and Lemma 2.5, has cokernel ik×{0}. Then, by a standard +projection argument onto some orthogonal complement of ik, we can apply the im- +plicit function theorem and obtain a C1 map (ε, b) �→ s(ε, b) such that �Ψ(b, ε, s(ε, b)) +lies in k, and conclude the proof by setting +ˇΦ(ε, b) = (Ab(ε, s(ε, b)))0,1 − A0,1. +□ +We will now explain that for each ε ∈ [0, ε0), the map +(3.2) +µε : +B +→ +k +b +�→ +ΛεiF ˇ +Aε,b − cε IdE +is a moment map for the K-action on B, for suitable symplectic forms Ωε on B. +Recall from [4, 11] that for ε ∈ (0, ε0), the gauge action of G C(π∗Gr(E), h) on +the affine space ∂0 + Ω0,1(X′, End(Gr(E))) is hamiltonian for the symplectic form +given, for (a, b) ∈ Ω0,1(X′, End(Gr(E)))2, by +(3.3) +ΩD +ε (a, b) = +� +X′ trace(a ∧ b∗) ∧ +ωn−1 +ε +(n − 1)!, +with equivariant moment map ∂ �→ ΛεFA∂ where A∂ stands for the Chern connec- +tion of (∂, h). Here, we identified the Lie algebra of G C(Gr(E), h) with its dual by +mean of the invariant pairing +(3.4) +⟨s1, s2⟩ε := +� +X′ trace(s1 · s∗ +2) ωn +ε +n! . +Note that the above expressions admit continuous extensions for ε = 0 when we +restrict to the G C(Gr(E), h0) action on ∂0 + Ω0,1(X, End(Gr(E))) and integrate +over (X, ω). + +14 +A. CLARKE AND C. TIPLER +Remark 3.5. We used above the Chern correspondence, for h fixed, between +Dolbeault operators and hermitian connections to express the infinite dimensional +moment map picture on the space of Dolbeault operators. +Proposition 3.6. Up to shrinking ε0 and B, for all ε ∈ [0, ε0), the map ∂0+ ˇΦ(ε, ·) +is a K-equivariant map from B to ∂0 + Ω0,1(X′, End(Gr(E))) whose image is a +symplectic submanifold for ΩD +ε . +Proof. The equivariance follows easily from Proposition 3.1 and from the construc- +tion of ˇΦ in the proof of Proposition 3.2. For ε = 0, the map ˇΦ(0, ·) is obtained by +perturbing �Φ = π∗ ◦ Φ. But Φ is complex analytic with, by construction, injective +differential at the origin (see e.g. the orginal proof [19] or [10]). So is �Φ, and thus +�Φ(B) is a complex subspace of Ω0,1(X′, End(π∗Gr(E))). We deduce that, up to +shrinking B, �Φ induces an embedding of B such that the restriction of ΩD +0 to �Φ(B) +is non-degenerate (recall that ΩD +0 is a K¨ahler form on the space of Dolbeault oper- +ators on X). As �Φ(ε, ·) is obtained by a small and continuous perturbation of �Φ, +and as being a symplectic embedding is and open condition, the result follows. +□ +From this result, we deduce that the map µε defined in (3.2) is a moment map +for the K-action on B with respect to the pulled back symplectic form +Ωε := ˇΦ(ε, ·)∗ΩD +ε , +and where we use the pairing ⟨·, ·⟩ε defined in (3.4) to identify k with its dual. From +the discussion of Section 3.1, E is obtained as a small complex deformation of Gr(E), +and thus by Proposition 3.1, ∂E is gauge equivalent to an element ∂b := ∂0 + Φ(b). +Then, from properties of the maps Φ and ˇΦ, for all ε ∈ [0, ε0) and for all g ∈ G, +π∗∂E will be gauge equivalent to π∗∂0 + ˇΦ(ε, g · b), provided g · b ∈ B. As a zero of +µε corresponds to a HYM connection on (X′, ωε), we are left with the problem of +finding a zero for µε in the G-orbit of b. +4. Proof of the main theorem +We carry on with notations from the last section, and our goal now is to prove +Theorem 1.1. This is where we will need to assume that in Gr(E) = �ℓ +i=1 Gi, all +stable components Gi are non isomorphic. This implies that +g = aut(Gr(E)) = +ℓ +� +i=1 +C · IdGi +and thus its compact form k is abelian, with K a compact torus. +4.1. The local convex cone associated to the K-action. In order to prove the +existence of a zero of µε in Z := G · b ∩ B, we start by describing, at least locally, +the images of Z by the maps (µε)ε∈[0,ε0). In this section, relying on [24], we will +see that those images all contain translations of (a neighbourhood of the apex of) +the same convex cone. +By simplicity of E, the stabiliser of b under the K-action is reduced to the S1- +action induced by gauge transformations of the form eiθ IdE. As those elements fix +all the points in B, elements in S1 · IdE will play no role in the arguments that +follow. Hence, we will work instead with the quotient torus K0 := K/S1 · IdE. +Note that the constants cε that appear in the maps µε in (3.2) are chosen so that +⟨µε, IdE⟩ε = 0. As the µε take vakues in k, this is equivalent to say trace(µε) = 0. + +BLOWING-UP HYM CONNECTIONS +15 +Hence, setting k0 ⊂ k to be the set of trace free elements in �ℓ +i=1 iR · IdGi, we will +consider the family of moment maps µε : B → k0 for the K0-action, and we may, +and will, assume that the stabiliser of b is trivial. Then, by using the inner product +⟨·, ·⟩ε to identify k0 ≃ k∗ +0, we can see the maps µε as taking values in k∗ +0 : +µ∗ +ε : B → k∗ +0. +There is a weight decomposition of V under the abelian K-action +(4.1) +V := +� +m∈M +Vm +for M ⊂ k∗ +0 the lattice of characters of K0. In the matrix blocks decomposition +of V = H0,1(X, End(Gr(E))) induced by Gr(E) = �ℓ +i=1 Gi, using the product +hermitian metric h0, we have +V = +� +1≤i,j≤ℓ +H0,1(X, G∗ +i ⊗ Gj). +The action of g ∈ K0 on Vij := H0,1(X, G∗ +i ⊗ Gj) is, by Equation (3.1): +(4.2) +g · γij = gig−1 +j γij. +Thus, in the weight space decomposition (4.1), Vij is the eigenspace with weight +(4.3) +mij := (0, . . . , 0, 1, 0, . . ., 0, −1, 0, . . ., 0) +where +1 appears in i-th position and −1 in the j-th position. If we decompose b +accordingly as +(4.4) +b = +� +ij +bij, +where bij ∈ Vij is non zero, as ∂E = ∂0 + γ with γ upper triangular, or equivalently +as E is obtained as successive extentions of the stable components Gi’s, only indices +(i, j) with i < j will appear in (4.4). From now on, we will restrict our setting to +B ∩ +� +bij̸=0 +Vij, +which we still denote by B. That is, we only consider weight spaces that appear in +the decomposition of b. Similarily, we use the notation V for � +bij̸=0 Vij. +To sum up, we are in the following setting : +(R1) The compact torus K0 acts effectively and holomorphically on the complex +vector space V ; +(R2) There is a continous family of symplectic forms (Ωε)0≤ε<ε0 on B ⊂ V +around the origin, with respect to which the K0-action is hamiltonian; +(R3) The point b ∈ B has trivial stabiliser, 0 in its KC +0 -orbit closure, and for all +weight mij ∈ M appearing in the weight space decomposition of V , bij ̸= 0. +(R4) The restriction of the symplectic form Ω0 to the KC +0 -orbit of b is non- +degenerate. +This last point follows as in the proof of Proposition 3.6. We set +Z := B ∩ (KC +0 · b). +We also introduce +σ := +� +bij̸=0 +R+ · mij ⊂ k∗ +0 + +16 +A. CLARKE AND C. TIPLER +with {mij, bij ̸= 0} the set of weights that appear in the decomposition of b ∈ V , +and for η > 0 +ση := +� +bij̸=0 +[0, η) · mij ⊂ k∗ +0. +Note that by the local version of Atiyah and Guillemin–Sternberg’s convexity the- +orem, there exists η > 0 such that µ∗ +ε(0) + ση ⊂ µ∗ +ε(B) for all ε small enough (see +the equivariant Darboux Theorem [14, Theorem 3.2] combined with the local de- +scription of linear hamiltonian torus actions [14, Section 7.1]). By [24, Proposition +4.6], the properties (R1) − (R4) listed above actually imply : +Proposition 4.1. Up to shrinking B and ε0, there exists η > 0 such that for all +ε ∈ [0, ε0), +µ∗ +ε(0) + Int(ση) ⊂ µ∗ +ε(Z) +and +µ∗ +ε(0) + ση ⊂ µ∗ +ε(Z). +Remark 4.2. The fact that the interior of µ∗ +ε(0) + ση is included in the image +of the KC +0 -orbit of b by µ∗ +ε is not stated explicitely in [24], but follows from the +discussion at the beginning of the proof of [24, Proposition 4.6]. +4.2. Solving the problem. From Proposition 4.1, to prove the existence of a +zero of µε in Z, it is enough to show that −µ∗ +ε(0) ∈ Int(ση), which reduces to +−µ∗ +ε(0) ∈ Int(σ) for small enough ε. Arguing as in [24, Lemma 4.8], σ and its dual +σ∨ := {v ∈ k0 | ⟨m, v⟩ ≥ 0 ∀m ∈ σ} +are strongly convex rationnal polyhedral cones of dimension ℓ − 1. Note that here +the pairing ⟨·, ·⟩ is the natural duality pairing. By duality, σ = (σ∨)∨, and we are +left with proving +−µ∗ +ε(0) ∈ Int((σ∨)∨). +The cone σ∨ can be written +σ∨ = +� +a∈A +R+ · va +for a finite set of generators {va}a∈A ⊂ k0. Hence, our goal now is to show that for +all a ∈ A, ⟨µ∗ +ε(0), va⟩ < 0, which by construction is equivalent to +(4.5) +⟨µε(0), va⟩ε < 0, +under the asumption that for any F ∈ E[ω], +(4.6) +µLε(F) < +ε→0 µLε(E). +We will then study in more details Equations (4.5) and (4.6). In order to simplify +the notations, in what follows, we will assume that all the stable components of +Gr(E) have rank one, so that trace(IdGi) = 1 for 1 ≤ i ≤ ℓ. The general case can +easily be adapted, and is left to the reader. + +BLOWING-UP HYM CONNECTIONS +17 +4.2.1. Condition (4.5) : generators of the dual cone. We will give here a more +precise form for the generators {va}a∈A of σ∨. Recall from [15, Section 1.2] the +method to find such generators : as σ is ℓ − 1-dimensional, each of its facets is +generated by ℓ − 2 elements amongst its generators (mij). Then, a generator va for +σ∨ will be an “inward pointing normal” to such a facet. Hence, if +va = +ℓ +� +i=1 +ai IdGi +is a generator of σ∨, there exists a set S := {mij} of ℓ − 2 generators of σ such that +∀ mij ∈ S, ⟨mij, va⟩ = 0. +Moreover, va ∈ k0 should be trace free, and as we assume here rank(Gi) = 1 for all +stable components, it gives +ℓ +� +i=1 +ai = 0. +Lemma 4.3. Up to scaling va, there exists a partition {1, . . ., ℓ} = I− ∪ I+ such +that for all i ∈ I−, ai = − +1 +♯I− and for all i ∈ I+, ai = +1 +♯I+ , where ♯ stands for the +cardinal of a set. +Proof. The key is to observe that if mij, mjk ∈ S2, then mik /∈ S. Indeed, by +(4.3), mij + mjk = mik, and those are generators of the cone. Equivalently, if +mij, mik ∈ S2, then mjk /∈ S. We then assign an oriented graph Ga to va. The +vertices are labelled a1 to aℓ, and we draw an oriented edge from ai to aj if ai = aj +and i < j. For each mij ∈ S, ⟨mij, va⟩ = 0 gives ai = aj. Hence, Ga has at least +ℓ − 2 edges. To prove the result, it is enough to show that Ga has 2 connected +components. Indeed, we can then set I− = {i |ai < 0} and I+ = {i |ai > 0}. All +elements ai for i ∈ I− will correspond to the same connected component and be +equal, and similarily with i ∈ I+. As �ℓ +i=1 ai = 0, we obtain the result by rescaling. +Proving that Ga has two connected components is then routine. It has ℓ vertices +and ℓ − 2 oriented edges, with the rule that if there is an edge from ai to aj and an +edge from ai to ak, then there is no edge from aj to ak. We consider the number of +edges that start from a1. If there are ℓ − 2 of those, then the connected component +of a1 has at least ℓ − 1 vertices, and we are left with at most 1 singleton for the +other component. The fact that va is trace free imposes that there are at least 2 +connected components, and we are done in that case. Then, if there are ℓ − 2 − k +edges from a1, its connected component has at least ℓ − 1 − k elements, and we are +left with at most k + 1 vertices and k edges for the other components. But its easy +to show, by induction on k, that the rule stated above implies that there will be at +most 1 connected component for such a graph with k + 1 vertices and k edges, and +we are done. +□ +We can now translate condition (4.5), by Lemma 4.3, it is equivalent to +(4.7) +� +i∈I+⟨µε(0), IdGi⟩ε +♯I+ +< +� +i∈I−⟨µε(0), IdGi⟩ε +♯I− +. + +18 +A. CLARKE AND C. TIPLER +4.2.2. Condition (4.6) : one parameter degenerations. We will associate to each +generator va of σ∨ a subsheaf F ∈ E[ω]. Geometrically, the idea is that va ∈ k0 +generates a one-parameter subgroup of K0 and a degeneration of E to F ⊕ E/F, +to which is assigned the Hilbert–Mumford weight µLε(F) − µLε(E) < 0. We let +va = �ℓ +i=1 ai IdGi ∈ σ∨ a generator as above, and define +Fa = +� +i∈I+ +Gi, +as a smooth complex vector bundle, and will show that ∂E(Fa) ⊂ Ω0,1(X′, Fa). This +implies that Fa ∈ E[ω] as a holomorphic vector bundle, with Dolbeault operator the +restriction of ∂E. Recall that ∂E = ∂0 + γ = ∂0 + � +bij̸=0 γij, that is, by choice of +b, the weights that appear in the weight decomposition of γ are the same as those +that appear in the decomposition of b. In the matrix blocks decomposition given +by �ℓ +i=1 Gi, the operator ∂0 is diagonal, and thus sends Fa to Ω0,1(X′, Fa). We +need to show that for each j ∈ I+, γ(Gj) ⊂ Ω0,1(X′, Fa). As va ∈ σ∨, it satisfies, +for all generator mij of σ : +⟨mij, va⟩ ≥ 0, +that is, for all (i, j) with i < j and bij ̸= 0, +ai − aj ≥ 0. +As j ∈ I+, this implies ai ≥ aj > 0. Hence, if i < j is such that bij ̸= 0, then +i ∈ I+. Equivalently, for i < j, i ∈ I− implies γij = 0, and thus we see that +γ(Gj) ⊂ Ω0,1(X′, Fa), and hence ∂E(Fa) ⊂ Ω0,1(X′, Fa). +Then we have Fa ∈ E[ω] and Condition (4.6) gives +µLε(Fa) < +ε→0 µLε(E), +which, by the see-saw property of slopes (see e.g. [25, Corollary 3.5] ), gives +µLε(Fa) < +ε→0 µLε(E/Fa) +and thus (recall we assume rank(Gi) = 1): +(4.8) +� +i∈I+ µLε(Gi) +♯I+ +< +ε→0 +� +i∈I− µLε(Gi) +♯I− +. +4.2.3. Conclusion. Recall that Equation (4.8) means that in the ε-expansion of +� +i∈I+ µLε (Gi) +♯I+ +− +� +i∈I− µLε(Gi) +♯I− +, the first non-zero term is strictly negative. By Chern– +Weyl theory, using the fact that A0 and ˇAε,0 are gauge-equivalent by point (2) of +Proposition 3.2, we have +µLε(Gi) += +c1(Gi) · [ωε]n−1 += +1 +2π⟨µε(0), IdGi⟩ε + cε +2π⟨IdE, IdGi⟩ε. +Hence Inequality (4.8) implies Inequality (4.7), for ε small enough, which concludes +the existence of bε ∈ Z such that µε(bε) = 0. Then, by construction, the associated +connections ˇAε,bε provide HYM connections with respect to ωε on bundles gauge +equivalent to E, where the gauge equivalences are given by elements in the finite +dimensional Lie group Aut(Gr(E)). To conclude the proof of Theorem 1.1, it then +remains to show that the connections ˇAε,bε converge to π∗A0 = ˇA0,0 in any L2,l +Sobolev norm. By construction of ˇAε,b in Proposition 3.2, it is enough to prove + +BLOWING-UP HYM CONNECTIONS +19 +that bε converges to 0 when ε → 0. Recall from [14, Theorem 3.2 and Section 7.1] +that B can be chosen so that µ∗ +ε is given by +(4.9) +µ∗ +ε(b′) = µ∗ +ε(0) + +� +ij +||b′ +ij||2 +ε · mij, +for some norm || · ||ε that depends continously on ε. As µε(0) → +ε→0 µ0(0) = 0, the +equation µ∗ +ε(bε) = 0 implies that for all (i, j), ||(bε)ij||ε → +ε→0 0. As the norms || · ||ε +vary continuously, they are mutually bounded, and thus bε → +ε→0 0, which concludes +proof of Theorem 1.1. +4.2.4. Proof of the corollaries. We comment now on the various corollaries stated in +the introduction. First, Corollary 1.3 is a direct application of Theorem 1.1, where +E = Gr(E) as a single stable component. Corollary 1.4 also follows directly, using +Formula (1.2). What remains is to show Corollary 1.5. The only remaing case to +study is when for all F ∈ E[ω], µLε(F) ≤ +ε→0 µLε(E), with at least one equality. In +that situation, the discussion in the last two sections shows that −µε(0) ∈ σ will lie +in the boundary of σ. Hence, by Proposition 4.1, there is a boundary point b′ ∈ Z in +the orbit closure of b with µε(b′) = 0. This point corresponds to a HYM connection +on a vector bundle that is then polystable for the holomorphic structure given by +ˇA0,1 +ε,b′, with respect to Lε. As this bundle correspond to a boundary point in the +complex orbit of b, it admits a small complex deformation to π∗E. As semi-stability +is an open condition, we deduce that π∗E is itself semi-stable for Lε. +References +[1] Claudio Arezzo and Frank Pacard. Blowing up and desingularizing constant scalar curvature +K¨ahler manifolds. Acta Math., 196(2):179–228, 2006. 1 +[2] Claudio Arezzo and Frank Pacard. Blowing up K¨ahler manifolds with constant scalar curva- +ture. II. Ann. of Math. (2), 170(2):685–738, 2009. 1, 3 +[3] Claudio Arezzo, Frank Pacard, and Michael Singer. Extremal metrics on blowups. Duke Math. +J., 157(1):1–51, 2011. 1, 3 +[4] M. F. Atiyah and R. Bott. The Yang-Mills equations over Riemann surfaces. Philos. Trans. +Roy. Soc. London Ser. A, 308(1505):523–615, 1983. 13 +[5] Nicholas Buchdahl and Georg Schumacher. Polystable bundles and representations of their +automorphisms. Complex Manifolds, 9(1):78–113, 2022. 9, 10, 12 +[6] Nicholas P. Buchdahl. Blowups and gauge fields. Pacific J. Math., 196(1):69–111, 2000. 1, 3 +[7] Ruadha´ı Dervan. Stability conditions for polarised varieties. ArXiv preprint arXiv:2103.03177, +2021. 11 +[8] Ruadha´ı Dervan and Lars Martin Sektnan. Extremal K¨ahler metrics on blow-ups. ArXiv +preprint arXiv:2110.13579, 2021. 1, 3, 11 +[9] Ruadha´ı Dervan and Lars Martin Sektnan. Hermitian Yang-Mills connections on blowups. J. +Geom. Anal., 31(1):516–542, 2021. 1, 3 +[10] A.-K. Doan. Group actions on local moduli space of holomorphic vector bundles. ArXiv +preprint arXiv:2201.10851, 2022. 10, 14 +[11] S. K. Donaldson. Anti self-dual Yang-Mills connections over complex algebraic surfaces and +stable vector bundles. Proc. London Math. Soc. (3), 50(1):1–26, 1985. 13 +[12] S. K. Donaldson. Infinite determinants, stable bundles and curvature. Duke Math. J., +54(1):231–247, 1987. 1, 5 +[13] Simon K. Donaldson. K¨ahler geometry on toric manifolds, and some other manifolds with +large symmetry. In Handbook of geometric analysis. No. 1, volume 7 of Adv. Lect. Math. +(ALM), pages 29–75. Int. Press, Somerville, MA, 2008. 9 + +20 +A. CLARKE AND C. TIPLER +[14] Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, and Theo van den Hurk. Hamiltonian +group actions and equivariant cohomology. SpringerBriefs in Mathematics. Springer, Cham, +2019. 16, 19 +[15] William Fulton. Introduction to toric varieties, volume 131 of Annals of Mathematics Stud- +ies. Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in +Geometry. 17 +[16] Daniel Huybrechts and Manfred Lehn. The geometry of moduli spaces of sheaves. Cambridge +Mathematical Library. Cambridge University Press, Cambridge, second edition, 2010. 4, 5 +[17] Shoshichi Kobayashi. Curvature and stability of vector bundles. Proc. Japan Acad. Ser. A +Math. Sci., 58(4):158–162, 1982. 1, 5 +[18] Shoshichi Kobayashi. Differential geometry of complex vector bundles. Princeton Legacy +Library. Princeton University Press, Princeton, NJ, [2014]. Reprint of the 1987 edition [ +MR0909698]. 4 +[19] M. Kuranishi. New proof for the existence of locally complete families of complex structures. +In Proc. Conf. Complex Analysis (Minneapolis, 1964), pages 142–154. Springer, Berlin, 1965. +10, 14 +[20] Martin L¨ubke. Stability of Einstein-Hermitian vector bundles. Manuscripta Math., 42(2- +3):245–257, 1983. 1, 5 +[21] V. B. Mehta and A. Ramanathan. Restriction of stable sheaves and representations of the +fundamental group. Invent. Math., 77(1):163–172, 1984. 3 +[22] David Mumford. Projective invariants of projective structures and applications. In Proc. +Internat. Congr. Mathematicians (Stockholm, 1962), pages 526–530. Inst. Mittag-Leffler, +Djursholm, 1963. 1, 5 +[23] Achim Napame and Carl Tipler. Toric sheaves, stability and fibrations. ArXiv preprint +arXiv:2210.04587, 2022. 2, 3 +[24] Lars Martin Sektnan and Carl Tipler. Analytic K-semi-stability and wall crossing. In prepa- +ration. 3, 11, 14, 16 +[25] Lars Martin Sektnan and Carl Tipler. Hermitian Yang–Mills connections on pullback bundles. +ArXiv preprint arXiv:2006.06453, 2020. 7, 8, 18 +[26] Reza Seyyedali and G´abor Sz´ekelyhidi. Extremal metrics on blowups along submanifolds. J. +Differential Geom., 114(1):171–192, 2020. 1, 3 +[27] G´abor Sz´ekelyhidi. The K¨ahler-Ricci flow and K-polystability. Amer. J. Math., 132(4):1077– +1090, 2010. 3, 9 +[28] G´abor Sz´ekelyhidi. Blowing up extremal K¨ahler manifolds II. Invent. Math., 200(3):925–977, +2015. 1, 3 +[29] Fumio Takemoto. Stable vector bundles on algebraic surfaces. Nagoya Math. J., 47:29–48, +1972. 1, 5 +[30] K. Uhlenbeck and S.-T. Yau. On the existence of Hermitian-Yang-Mills connections in stable +vector bundles. volume 39, pages S257–S293. 1986. Frontiers of the mathematical sciences: +1985 (New York, 1985). 1, 5 +[31] Claire Voisin. Hodge theory and complex algebraic geometry. I, volume 76 of Cambridge +Studies in Advanced Mathematics. Cambridge University Press, Cambridge, english edition, +2007. Translated from the French by Leila Schneps. 7 +Andrew Clarke, Instituto de Matem´atica, Universidade Federal do Rio de Janeiro, +Av. Athos da Silveira Ramos 149, Rio de Janeiro, RJ, 21941-909, Brazil +Email address: andrew@im.ufrj.br +Carl Tipler, Univ Brest, UMR CNRS 6205, Laboratoire de Math´ematiques de Bre- +tagne Atlantique, France +Email address: Carl.Tipler@univ-brest.fr + diff --git a/5tAyT4oBgHgl3EQfpfgu/content/tmp_files/load_file.txt b/5tAyT4oBgHgl3EQfpfgu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..30e1121198855a73f5590eb5b27b39455a2e6977 --- /dev/null +++ b/5tAyT4oBgHgl3EQfpfgu/content/tmp_files/load_file.txt @@ -0,0 +1,804 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf,len=803 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='00525v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='DG] 2 Jan 2023 BLOWING-UP HERMITIAN YANG–MILLS CONNECTIONS ANDREW CLARKE AND CARL TIPLER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We investigate hermitian Yang–Mills connections for pullback vec- tor bundles on blow-ups of K¨ahler manifolds along submanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Under some mild asumptions on the graded object of a simple and semi-stable vector bun- dle, we provide a necessary and sufficent numerical criterion for the pullback bundle to admit a sequence of hermitian Yang–Mills connections for polarisa- tions that make the exceptional divisor sufficiently small, and show that those connections converge to the pulled back hermitian Yang-Mills connection of the graded object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Introduction A corner stone in gauge theory is the Hitchin–Kobayashi correspondence ([17, 20, 30, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' This celebrated generalisation of the Narasimhan and Seshadri the- orem asserts that a holomorphic vector bundle over a K¨ahler manifold carries an Hermite–Einstein metric if and only if it is polystable in the sense of Mumford and Takemoto ([22, 29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The interplay between the differential geometric side, her- mitian Yang–Mills connections (HYM for short) that originated from physics, and the algebro-geometric side, the stability notion motivated by moduli constructions, has had many applications and became a very fertile source of inspiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Given that HYM connections are canonically attached to polystable vector bundles, it is natural to investigate their relations to natural maps between vector bundles, such as pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In this paper, we address the problem of pulling back HYM connec- tions along blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' While the similar problem for extremal K¨ahler metrics has seen many developments in the past ten years [1, 2, 3, 28, 26, 8], relatively little seems to be known about the behaviour of HYM connections under blow-ups [6, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In this paper, under some mild asumptions, we solve the problem for pullback of semi-stable vector bundles on blow-ups along smooth centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Let π : X′ → X be the blow-up of a polarised K¨ahler manifold (X, [ω]) along a submanifold Z ⊂ X, and E′ = π∗E the pullback of a holomorphic vector bundle E → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For 0 < ε ≪ 1, Lε := π∗[ω] − εc1(Z′) defines a polarisation on X′, where we set Z′ = π−1(Z) the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' There are obstructions for E′ to admit HYM connections with respect to ωε ∈ c1(Lε), with 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In particular, E should be simple and semi-stable with respect to [ω] (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In the latter case, E admits a Jordan–Holder filtration by semi-stable sheaves with polystable graded object Gr(E) (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='2 for definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' A further obstruction comes then from subsheaves of E arising from Gr(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' While those sheaves have the same slope as E, their pullbacks to X′ could destabilise E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Our main result asserts that those are actually the only obstructions for E′ to carry a HYM connection, under some mild asumptions on Gr(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Primary: 53C07, Secondary: 53C55, 14J60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' 1 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' CLARKE AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' TIPLER Recall that a semi-stable holomorphic vector bundle E → (X, [ω]) is said to be sufficiently smooth if its graded object Gr(E) is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Let E[ω] denote the set of all subbundles of E arising in a Jordan–Holder filtration for E, or equivalently of same slope as E with respect to [ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For F ∈ E[ω], denote by µLε(F) = c1(π∗F)·Ln−1 ε rank(F) the slope of π∗F on (X′, Lε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Let E → X be a simple sufficiently smooth semi-stable holomorphic vector bundle on (X, [ω]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Assume that the stable components of Gr(E) are pairwise non-isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Then, there exists ε0 > 0 and a sequence of HYM connections (Aε)ε∈(0,ε0) on π∗E with respect to (ωε)ε∈(0,ε0) if and only if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1) ∀ F ∈ E[ω], µLε(F) < ε→0 µLε(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In that case, if A denotes a HYM connection on Gr(E) with respect to ω, then (Aε)ε∈(0,ε0) can be chosen so that Aε −→ ε→0 π∗A in any Sobolev norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In the statement, the expression µLε(F) < ε→0 µLε(E) means that the first non- zero term in the ε-expansion for µLε(E) − µLε(F) is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Simplicity, semi-stability and condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1) are necessary to pro- duce the connections (Aε) from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The other two asumptions on Gr(E) are technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Assuming Gr(E) to be locally free enables to see E as a smooth com- plex deformation of Gr(E) and to work with the various connections on the same underlying complex vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We should warn the reader though that if one drops this asumption, Condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1) might not be enough to ensure semi-stability of π∗E on (X′, Lε) (see the extra conditions in [23, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' On the other hand, the asumption on Gr(E) having no pairwise isomorphic components is purely technical, and ensures that its automorphism group, that will provide obstructions in the perturbative theory, is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We now list some corollaries of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' First, the stable case : Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Let E → X be a stable holomorphic vector bundle on (X, [ω]) and let A be a HYM connection on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Then, there exists ε0 > 0 and a sequence of HYM connections (Aε)ε∈(0,ε0) on π∗E with respect to (ωε)ε∈(0,ε0) such that Aε → ε→0 π∗A in any Sobolev norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For the semi-stable case, Condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1) reduces to a finite number of intersec- tion product computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' One interesting feature comes from the second term in the expansion of µLε(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' It is the opposite of the slope of the restriction of E to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The following formula is proved in [23, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1], where m = dim(Z) : (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='2) µLε(E) = µL(E) − �n − 1 m − 1 � µL|Z(E|Z)εn−m + O(εn−m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We then have : Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Let E → X be a simple sufficiently smooth semi-stable holomorphic vector bundle on (X, [ω]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Assume that the stable components of Gr(E) are pairwise non-isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Denote by A an HYM connection on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' If (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3) ∀ F ∈ E[ω], µL|Z(E|Z) < µL|Z(F|Z), then, there exists ε0 > 0 and a sequence of HYM connections (Aε)ε∈(0,ε0) on π∗E with respect to (ωε)ε∈(0,ε0) converging to π∗A in any Sobolev norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' BLOWING-UP HYM CONNECTIONS 3 Condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3) was checked on explicit examples in [23, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='5] to produce stable perturbations of tangent sheaves by blow-ups, and our result provides infor- mation on the associated connections and their asymptotic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Note that by Mehta–Ramanathan theorem [21], if [ω] = c1(L) is integral, and if Z is a generic intersection of divisors in linear systems |Lk|, then E|Z is semi-stable as soon as E is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In that case, Condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3) cannot be satisfied, and it seems unlikely that Con- dition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1) will hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Hence, blowing-up such subvarieties tend to destabilise a semi-stable bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In general, we expect that it should not be too hard to obtain stability of suffi- ciently smooth pulled back bundles under condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1) with purely algebraic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' However, we emphasize that the Hitchin–Kobayashi correspondence doesn’t provide any information on the asymptotic behaviour of the associated HYM connections, which is then the main content of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Nevertheless, we state the following corollary, that extends [23, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='10] to a non-equivariant situation: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Let E → X be a simple sufficiently smooth semi-stable holomorphic vector bundle on (X, [ω]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Assume that the stable components of Gr(E) are pairwise non-isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Then, there exists ε0 > 0 such that π∗E → (X′, Lε) is (i) stable if and only if for all F ∈ E[ω], µLε(F) < ε→0 µLε(E), (ii) semi-stable if and only if for all F ∈ E[ω], µLε(F) ≤ ε→0 µLε(E), (iii) unstable otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Finally, we comment on previous related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1 extends results from [6, 9] where blow-ups of HYM connections along points are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In the present paper, we consider blow-ups along any smooth subvariety, and also cover the semi-stable situation, which is technically more involved due to the presence of automorphisms of the graded object that obstruct the linear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' While [9] is a gluing construction as in the similar problem of producing extremal K¨ahler metrics on blow-ups [2, 3, 28, 26, 8], one of the key feature in our approach is to apply directly the implicit function theorem to reduce to (an ε dependent family of) finite dimensional GIT problems on a Kuranishi space parametrising small deformations of Gr(E), as in [27, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We then use the new technology developed in [24] to control the perturbations of the associated moment maps when ωε varries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' This is where our hypothesis on Aut(Gr(E)) being abelian is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The main new technical input comes from the fact that the underlying smooth manifold X is fixed in [24], while it varries with the blow-up, which requires a carefull analysis of the operator introduced to apply the implicit function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Outline: In Section 2, we recall basic material about HYM connections and stabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We then perform in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3 the analysis of the linear theory on the blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Relying on this, in Section 3 we explain how to reduce the problem to finding zeros of finite dimensional moment maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Then, we conclude the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1 and its corollaries in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Acknowledgments: The authors benefited from visits to LMBA and Gotheborg University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' they would like to thank these welcoming institutions for providing stimulating work environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The idea of this project emerged from discussions with Lars Martin Sektnan, whom we thank for sharing his ideas and insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' CT 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' CLARKE AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' TIPLER is partially supported by the grants MARGE ANR-21-CE40-0011 and BRIDGES ANR–FAPESP ANR-21-CE40-0017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Preliminaries In Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='2 we introduce the notions of HYM connections and slope stability, together with some general results, and refer the reader to [18] and [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' From Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3 we start to specialise the discussion to blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In particular, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='2, we provide various asymptotic expressions for the linearisation of the HYM equation on the blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Those results will be used in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The hermitian Yang–Mills equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Let E → X be a holomorphic vector bundle over a compact K¨ahler manifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' A hermitian metric on E is Hermite– Einstein with respect to a K¨ahler metric with K¨ahler form ω if the curvature Fh ∈ Ω2 (X, End E) of the corresponding Chern connection satisfies Λω (iFh) = c IdE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1) for some real constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Equivalently, if h is some hermitian metric on the smooth complex vector bundle underlying E, a hermitian connection A on (E, h) is said to be hermitian Yang–Mills if it satisfies � F 0,2 A = 0, Λω (iFA) = c IdE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The first equation of this system implies that the (0, 1)-part of A determines a holomorphic structure on E, while the second that h is Hermite–Einstein for this holomorphic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We will try to find hermitian Yang–Mills connections within the complex gauge group orbit, which we now define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The (hermitian) complex gauge group is G C(E, h) = Γ (GL (E, C)) ∩ Γ (EndH(E, h)) , where EndH(E, h) stands for the hermitian endomorphisms of (E, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Note that if ¯∂ is the Dolbeault operator defining the holomorphic structure on E, then f ◦ ¯∂◦f −1 defines a biholomorphic complex structure on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Let dA = ∂A + ¯∂A be the Chern connection of (E, h) with respect to the original complex structure (that is ¯∂A = ¯∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Then the Chern connection Af of h with respect to f ◦ ¯∂ ◦ f −1 is dAf = (f ∗)−1 ◦ ∂A ◦ (f ∗) + f ◦ ¯∂ ◦ f −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Solving the hermitian Yang–Mills equation is equivalent to solving Ψ(s) = c IdE where Ψ : Lie(G C(E, h)) −→ Lie(G C(E, h)) s �−→ iΛω(FAexp(s)), and where Lie(G C(E, h)) := iΓ(EndH(E, h)) is the tangent space to G C(E, h) at the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For a connection A on E, the Laplace operator ∆A is ∆A = iΛω �¯∂A∂A − ∂A ¯∂A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='2) If AEnd E denote the connection induced by A on End E, then : BLOWING-UP HYM CONNECTIONS 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' If A is the Chern connection of (E, ∂, h), the differential of Ψ at identity is dΨIdE = ∆AEnd E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' If moreover A is assumed to be hermitian Yang–Mills, then the kernel of ∆AEnd E acting on Γ(End(E)) is given by the Lie algebra aut(E) of the space of automor- phisms Aut(E) of (E, ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The last statement about the kernel follows from the K¨ahler identities and the Akizuki-Nakano identity that imply ∆AEnd E = ∂∗ A∂A + ¯∂∗ A ¯∂A, the two terms of which are equal if A is Hermitian Yang-Mills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The operator ∆AEnd E being elliptic and self-adjoint, aut(E) will then appear as a cokernel in the linear theory for perturbations of hermitian Yang–Mills connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Slope stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We recall some basic facts about slope stability, as intro- duced by [22, 29], and refer the interested reader to [16] for a detailed treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We denote here L := [ω] the polarisation of the n-dimensional K¨ahler manifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For E a torsion-free coherent sheaf on X, the slope µL(E) ∈ Q (with respect to L) is given by the intersection formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3) µL(E) = degL(E) rank(E) , where rank(E) denotes the rank of E while degL(E) = c1(E) · Ln−1 stands for its degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Then, E is said to be slope semi-stable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' slope stable) with respect to L if for any coherent subsheaf F of E with 0 < rank(F) < rank(E), one has µL(F) ≤ µL(E) ( resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' µL(F) < µL(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' A direct sum of slope stable sheaves of the same slope is said to be slope polystable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In this paper, we will often omit “slope” and simply refer to stability of a sheaf, the polarisation being implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We will make the standard identification of a holo- morphic vector bundle E with its sheaf of sections, and thus talk about slope sta- bility notions for vector bundles as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In that case slope stability relates nicely to differential geometry via the Hitchin–Kobayashi correspondence : Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3 ([17, 20, 30, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' There exists a Hermite–Einstein metric on E with respect to ω if and only if E is polystable with respect to L We will be mostly interested in semi-stable vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' A Jordan–H¨older filtration for a torsion-free sheaf E is a filtration by coherent subsheaves: 0 = F0 ⊂ F1 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' ⊂ Fℓ = E, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='4) such that the corresponding quotients, Gi = Fi Fi−1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='5) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' , ℓ, are stable with slope µL(Gi) = µL(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In particular, the graded object of this filtration (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='6) Gr(E) := l � i=1 Gi is polystable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' From [16, Section 1], we have the standard existence and uniqueness result: 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' CLARKE AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' TIPLER Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Any semi-stable coherent torsion-free sheaf E on (X, L) admits a Jordan–H¨older filtration, and the graded object Gr(E) of such filtrations is unique up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' When E is locally-free and semi-stable, we say that it is sufficiently smooth if Gr(E) is locally-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In that case, we denote E[ω] the set of holomorphic subbundles of E built out of successive extensions of some of the stable components of Gr(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Equivalently, E[ω] is the set of holomorphic subbundles of E arising in a Jordan- Holder filtration for E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Finally, we recall that a necessary condition for E to be stable is simplicity, that is Aut(E) = C∗ · IdE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Geometry of the blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We consider now Z ⊂ X a m-dimensional com- plex submanifold of codimension r = n − m ≥ 2 and the blow-up map π : BlZ(X) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We will denote by X′ = BlZ(X) the blown-up manifold and by Z′ = π−1(Z) the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We denote by Lε := π∗L − ε[Z′] a polarisation on X′, for 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Let E → X be a holomorphic vector bundle, and denote by E′ = π∗E the pulled back bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For any holomorphic subbundle F ⊂ E, the intersection numbers µLε(π∗E)−µLε(π∗F) admit expansions in ε, with first term given by µL(E) − µL(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For that reason, given the Hitchin–Kobayashi correspondence in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3, semi-stability of E on (X, L) is a necessary con- dition for its pullback E′ to admit an HYM connection with respect to a K¨ahler metric in Lε, for all 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Another necessary condition is simplicity of E′, which, by Hartogs’ theorem, is equivalent to simplicity of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Then, natural can- didates to test for stability of E′ are given by the pullbacks of elements in E[ω], and Condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1) clearly is necessary for E′ to be stable in the polarisations we consider, and thus to admit an HYM connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Hence, we will assume E to be simple, semi-stable, and to satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We now turn back to the differential geometry of the blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Decomposition on spaces of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We have a commutative diagramm: Z′ ι −→ X′ ↓ ↓ Z ι0 −→ X where ι0 and ι denote the inclusions, while the vertical arrows are given by the projection map π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We then have a pullback map on sections π∗ : Γ(X, End(E)) −→ Γ(X′, End(π∗E)) as well as a restriction map : ι∗ : Γ(X′, End(π∗E)) −→ Γ(Z′, End(ι∗π∗E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Our goal now is to fit those maps in a short exact sequence, that will in the end split the space Γ(X′, End(π∗E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' If NZ = T X|Z/T Z denotes the normal bundle of Z in X, then Z′ ≃ P(NZ), and we can fix a (1, 1)-form λ ∈ c1(OP(NZ)(1)) that restricts to K¨ahler metrics on the fibers of P(NZ) → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We also fix a K¨ahler form ω ∈ c1(L) on X, and consider its restriction to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We then have a K¨ahler CPr−1-fibration : π : (Z′, λ) −→ (Z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' BLOWING-UP HYM CONNECTIONS 7 By averaging along fibers as described in [25, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3], we obtain a splitting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='7) Γ(Z′, End(ι∗π∗E)) = π∗(Γ(Z, End(ι∗ 0E))) ⊕ Γ0(Z′, End(ι∗π∗E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We will omit the ι∗ and π∗ to simplify notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Using the projection on the second factor p0 : Γ(Z′, End(E)) → Γ0(Z′, End(E)) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='7), we deduce a short exact sequence : 0 −→ Γ(X, End(E)) π∗ −→ Γ(X′, End(E)) p0◦ι∗ −→ Γ0(Z′, End(E)) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We can actually split this sequence by mean of a linear extension operator ι∗ : Γ0(Z′, End(E)) −→ Γ(X′, End(E)) such that p0 ◦ ι∗ ◦ ι∗ = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' This can be done using bump functions and a standard partition of unity argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The outcome is an isomorphism : (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='8) Γ(X′, End(E)) −→ Γ(X, End(E)) ⊕ Γ0(Z′, End(E)) s �−→ (s − ι∗ ◦ p0 ◦ ι∗s , p0 ◦ ι∗s), with inverse map (sX, sZ) �→ (π∗sX + ι∗sZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' This splits the Lie algebra of gauge transformations, and will be used to identify contributions coming from X and from Z′ in the ε-expansion of the linearisation, which we describe in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' From now on, by abuse of notations, we will consider the spaces Γ(X, End(E)) and Γ0(Z′, End(E)) as subspaces of Γ(X′, End(π∗E)), and denote s = sX + sZ the decomposition of an element s ∈ Γ(X′, End(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Decomposition of the Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We extend λ to a closed (1, 1)-form over X′ as in [31, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='3] and consider the family of K¨ahler metrics on X′: ωε = π∗ω + ελ ∈ c1(Lε), 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Let A be a Hermitian connection on E, which we pull back to X′ and extend to the bundle End(π∗E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We will now study the Laplace operator ∆εs = iΛε(¯∂A∂A − ∂A ¯∂A)s acting on the various components of s = sX + sZ ∈ Γ(X′, End(E)), where Λε is the Lefschetz operator for the metric ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For this, we need to introduce an elliptic operator on Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The vertical Laplace operator, denoted ∆V : Γ0 (Z′, End(E)) → Γ0 (Z′, End(E)) , is the operator defined by the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Let σ ∈ Γ0(Z′, End(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Over a point x ∈ Z, take the restriction σx of σ to Z′ x = π−1(x), and consider σx as a map to Cp with components σi x in a trivialisation π∗ End(E)x ∼= Cp of the restriction of π∗ End(E) to the fibre Z′ x of Z′ → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Define (∆V (σ))i x = ∆(λ)|Z′x � σi x � , for ∆λ the Laplacian of the K¨ahler form λ on Z′ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Then glue together to form a section of π∗ End(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' As in [25, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1], one easily obtains that this con- struction is independent on the trivialisation chosen, and sends smooth sections to smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In the following Lemma, the supscript l (or l + 2) stands for the Sobolev completion with respect to some L2,l Sobolev norm, where those norms 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' CLARKE AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' TIPLER can be produced out of the metrics ω, λ and any metric h on E, together with the covariant derivatives given by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' [25, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1] The vertical Laplacian ∆V : Γ0 (Z′, End(E))l+2 → Γ0 (Z′, End(E))l is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In the following statements, if A denotes a second order operator acting on sections, then in an expression of the form A(σ) = σ0 + εσ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' + εd−1σd−1 + O(εd) the term O(εd) will stand for σd ·εd, where σd is a section whose L2,l Sobolev norm is bounded by the L2,l+2 Sobolev norm of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' If sZ = ι∗σZ for σZ ∈ Γ(Z′, End(E)), then (p0 ◦ ι∗)∆ε(ι∗σZ) = ε−1∆VσZ + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We introduce the operator D given by DsZ = i(¯∂A∂A − ∂A ¯∂A)sZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The Laplacian ∆ε satisfies on X′ : ∆εsZ ωn ε = nDsZ ∧ ωn−1 ε , or equivalently ∆εsZ = n DsZ ∧ (ω + ελ)n−1 (ω + ελ)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We note that ω is a K¨ahler form on X, but on X′ is degenerate along the fibre directions of the submanifold Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Then (i∗ω)m+1 = 0 ∈ Ω2(m+1)(Z′), and at x ∈ Z′ ⊆ X′, ωm+2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Then, expanding (ω + ελ)n−1 and (ω + ελ)n gives ι∗∆εsZ = (n − m − 1)ε−1 DsZ ∧ ωm+1 ∧ λn−m−2 ωm+1 ∧ λn−m−1 + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Restricting to Z′, the connection 1-forms of A vanish, so ι∗DsZ = i∂ ¯∂σZ, acting on the coefficient functions of σZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' On the other hand, by considering a convenient orthonormal frame at x ∈ Z′, we see that ι∗∆ει∗σZ = ε−1∆VσZ + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' □ In the next lemma, we denote ∆εsZ = (∆εsZ)X + (∆εsZ)Z the decomposition according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For sZ = ι∗σZ with σZ ∈ Γ(Z′, End(E)), we have (∆εsZ)X = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' By definition, (∆εsZ)X = π∗φ for some φ ∈ Γ(X, End(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' As we also have (∆εsZ)X = (Id − ι∗(p0 ◦ ι∗))ΛεDsZ, we deduce that the section φ is the continuous extension of π∗(Id−ι∗(p0◦ι∗))ΛεDsZ across Z ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' On X′ \\ Z′ we have ΛεDsZ = nDsZ ∧ (ωn−1 + O(ε)) ωn + O(ε) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' As π∗(Id − ι∗(p0 ◦ ι∗)) is O(1), the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' □ BLOWING-UP HYM CONNECTIONS 9 From the previous two lemmas, in the decomposition s = sX + sZ, ∆εsZ also lies in the subspace Γ0(Z′, End(E)) ⊆ Γ(X′, End(E)) to higher order in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For sX ∈ Γ(X, End(E)), ∆εsX = (∆εsX)X + (∆εsX)Z where (∆εsX)Z = ι∗(p0 ◦ ι∗)∆εsX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We first consider ι∗∆εsX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For sX = π∗σX ∈ Γ(X, End(E)) ⊆ Γ(X′, End(E)), ι∗∆εsX = (m + 1)DsX ∧ ωm ∧ λn−m−1 ωm+1 ∧ λn−m−1 + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Firstly, sX = π∗σX, and the connection A is pulled back from X, so DsX is basic for the projection to X and Ds ∧ ωm+1 = 0 at points in Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Secondly, we note that ωm+1 ∧λn−m−1 is a volume form on X′, in a neighbourhood of Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Then, the result follows similarly to the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' □ For the final term (∆εsX)X, we introduce ∆X the Laplace operator of A on End(E) → (X, ω): ∆X : Γ (X, End(E)) → Γ (X, End(E)) σ �→ iΛω(¯∂A∂A − ∂A ¯∂A)σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' For sX = π∗σX ∈ Γ(X, End(E)) ⊆ Γ(X′, End(E)), (∆εsX)X = π∗(∆XσX) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' There is φ ∈ Γ(X, End(E)) such that (∆εsX)X = π∗φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The element φ can be identified as the lowest order term in the asymptotic expansion in ε of (∆επ∗σX)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' However, we have at x ∈ X′ \\ Z′ : ∆επ∗σX = nDπ∗σX ∧ (ω + ελ)n−1 (ω + ελ)n = nπ∗ DσX ∧ ωn−1 ωn + O(ε) so we see that the lowest order term in the expansion of (∆επ∗σX)X is ∆XσX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' □ Summarizing the above calculations, with respect to the decomposition s = sX + sZ produced by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='8), the operator ∆ε takes the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='9) � ∆X 0 L ε−1∆V � plus higher order terms, for some second order operator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' In the next section, we will apply the previous lemmas and the resulting form for ∆ε to the pullback of a HYM connection A0 on the graded object Gr(E) of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The perturbation argument The goal of this section is to reduce the problem of finding a zero for the operator s �→ iΛωε(FAexp(s)) − cεId in a gauge group orbit to a finite dimensional problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The ideas here go back to [13, 27], and our framework will be that of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' CLARKE AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' TIPLER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Kuranishi slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We start from a simple semi-stable and sufficiently smooth holomorphic vector bundle E on (X, L), with L = [ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' Denote by Gr(E) = �ℓ i=1 Gi the associated polystable graded object, with stable components Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' We let ∂0 be the Dolbeault operator of Gr(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The automorphism group G := Aut(Gr(E)) is a reductive Lie group with Lie algebra g := aut(Gr(E)) and compact form K ⊂ G, with k := Lie(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfpfgu/content/2301.00525v1.pdf'} +page_content=' The Dolbeault operator ∂E on E is given by ∂E = ∂0 + γ where γ ∈ Ω0,1(X, Gr(E)∗ ⊗ Gr(E)) can be written γ = � i