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+{"text":"\\section{Introduction}\n Chinese is a  logographic language. \nThe logograms of Chinese characters encode rich information of their meanings. Therefore, it is intuitive that\nNLP tasks for  Chinese should benefit from the use of the glyph information.\nTaking \ninto account logographic information should help semantic modeling. \nRecent studies indirectly support this argument: \nRadical representations  have proved to be useful in a wide range of language understanding tasks \\citep{shi2015radical,li2015component,yin2016multi,sun2014radical,shao2017character}.\nUsing the Wubi  scheme ---- a Chinese character encoding method that mimics the order of typing the sequence of radicals for a character on the computer keyboard ---- is reported to  improve  performances on \nChinese-English\nmachine translation  \\citep{tan2018wubi2en}. \\citet{cao2018cw2vec} gets down to units of greater granularity, and proposed  stroke n-grams for character modeling.  \n\n\n\\begin{figure}\n\\center\n\\includegraphics[width=2.5in]{newfigs\/1.png}\n\\includegraphics[width=2.5in]{newfigs\/2.png}\n\\centering\n\\caption{Many Chinese characters evolved from pictures. The image is copied from Wikipedia.}\n\\label{illustration}\n\\end{figure}\n\n\n\n\n\nRecently, there have been some efforts \napplying CNN-based algorithms on the visual features of characters.\nUnfortunately, they do not show consistent performance boosts \\citep{liu2017learning,zhang2017encoding}, and some even yield negative results \\citep{dai2017glyph}. \nFor instance, \\citet{dai2017glyph} \nrun CNNs on char logos\nto obtain Chinese character representations and used them in the \ndownstream\nlanguage modeling task. They reported that   the incorporation of glyph representations actually worsen the performance and concluded that CNN-based representations do not provide extra useful information for language modeling.\nUsing similar strategies, \n\\citet{liu2017learning} and \\citet{zhang2017encoding} test the idea on text classification tasks, and performance boosts were observed only in very limited number of settings.\nPositive results come from \\citet{su2017learning}, which found glyph embeddings help two tasks:\nword analogy and word similarity. Unfortunately,  \\citet{su2017learning} only focus on word-level semantic tasks and\ndo not extend improvements in the word-level tasks\n to higher level NLP tasks such as phrase, sentence or discourse level.  \n Combined with radical representations, \n\\citet{shao2017character} run CNNs on character figures and \nuse the output as auxiliary features in the POS tagging task. \n\n We propose the following explanations for negative results reported in the earlier CNN-based models \\citep{dai2017glyph}:\n (1) not using the correct version(s) of script: Chinese character system has a long history of evolution.\nThe characters started from being easy-to-draw, and slowly transitioned to being easy-to-write. Also, they became less pictographic and less concrete over time. \nThe most widely used script version to date, the {\\it Simplified Chinese}, is the easiest script to write, but  inevitably loses the most significant amount of pictographic information. \n For example,  \"\u4eba\" (human) and  \"\u5165\" (enter), which are of irrelevant meanings, are highly similar in shape in simplified Chinese, but very different in historical languages such as bronzeware script. \n  (2) not using the proper CNN structures: \nunlike  ImageNet images \\citep{deng2009imagenet}, the size of which is mostly at the scale of\n800*600, \ncharacter logos are significantly smaller (usually with the size of 12*12). \nIt\n requires a different CNN architecture to capture the local graphic features of character images;\n(3) no regulatory functions were used in previous work: \nunlike the classification task on the imageNet dataset, which contains tens of millions of data points, \nthere are only about 10,000 Chinese characters. Auxiliary training objectives are thus critical in preventing overfitting and  \npromoting the model's ability to generalize.\n\nIn this paper, we propose the GLYCE, the GLYph-vectors for Chinese character representations. We treat Chinese characters as images and use CNNs to obtain their representations. \nWe resolve the aforementioned issues by using the following key techniques:\n\\begin{tightitemize}\n\\item We use the ensemble of the historical and the contemporary scripts (e.g., the bronzeware script, the clerical script, the seal script, the traditional Chinese etc), along with the scripts of different writing styles (e.g, the cursive script) to enrich pictographic information from the character images.\n\\item We utilize the Tianzige-CNN  (\u7530\u5b57\u683c) structures tailored to logographic character modeling for Chinese.\n\\item We use multi-task learning methods by adding an image-classification loss function \n to increase the model's ability to generalize.\n\\end{tightitemize}\nGlyce is found to improve a wide range of Chinese NLP tasks.\nWe are able\nto obtain the SOTA performances on a wide range of Chinese NLP tasks, including\n tagging (NER, CWS, POS), \nsentence pair classification (BQ, LCQMC,  XNLI, NLPCC-DBQA), \nsingle sentence classification tasks (ChnSentiCorp, the Fudan corpus, iFeng),\ndependency parsing, and semantic role labeling.\n\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=6in]{newfigs\/CNN-TIANZIGE.png}\n\\centering\n\\caption{Illustration of the Tianzege-CNN used in Glyce. }\n\\label{TIANZIGE}\n\\end{figure*}\n\n\n\n \n\n\\section{Glyce}\n\\begin{table*}\n\\small\n\\center\n\\begin{tabular}{|c|c|c|}\\hline\nChinese & English & Time Period \\\\\\hline\n\u91d1\u6587&Bronzeware script& Shang dynasty and Zhou dynasty (2000 BC \u2013 300 BC) \\\\\\hline\n\u96b6\u4e66& Clerical script & Han dynasty (200BC-200AD)  \\\\\\hline\n\u7bc6\u4e66& Seal script & Han dynasty and Wei-Jin period (100BC - 420 AD) \\\\\\hline\n\u9b4f\u7891& Tablet script & Northern and Southern dynasties 420AD - 588AD \\\\\\hline\n\\multirow{2}{*}{\u7e41\u4f53\u4e2d\u6587} & \\multirow{2}{*}{Traditional Chinese} & 600AD - 1950AD (mainland China).  \\\\\n& & still currently used in HongKong and Taiwan  \\\\\\hline\n\u7b80\u4f53\u4e2d\u6587 (\u5b8b\u4f53) &Simplified Chinese - Song & 1950-now \\\\\\hline\n\u7b80\u4f53\u4e2d\u6587 (\u4eff\u5b8b\u4f53) & Simplified Chinese - FangSong & 1950-now \\\\\\hline\n\u8349\u4e66&Cursive script& Jin Dynasty  to now  \\\\\\hline\n\\end{tabular}\n\\caption{Scripts and writing styles used in Glyce.}\n\\label{script}\n\\end{table*}\n\n\\subsection{Using Historical Scripts}\nAs discussed in Section 1, pictographic \ninformation is heavily lost in the simplified Chinese script. \nWe  thus propose using scripts from various time periods in history and also of different writing styles.\nWe collect the following major historical script with details  shown in Table \\ref{script}.\n Scripts from different historical periods, which are usually very different in shape, help the model to integrate \npictographic\nevidence from various sources;\nScripts of different writing styles  help improve the model's ability to generalize. \nBoth strategies are akin to widely-used data augmentation strategies in computer vision. \n\n\n\n\n\\subsection{The Tianzige-CNN Structure for Glyce}\n\\begin{comment}\n\\begin{table}\n\\center\n\\small\n\\begin{tabular}{ |c|c|c| }\\hline\n layer & kernel size & feature size\\\\ \n input &  & n$\\times$ 12$\\times$ 12 \\\\\n conv & 5 & 1024$\\times$8$\\times$8 \\\\ \n relu &  & 1024$\\times$8$\\times$8 \\\\ \n maxpool & 4 & 1024$\\times$ 2$\\times$2 \\\\\n 8-group conv& 1 & 256$\\times$2$\\times$ 2 \\\\\n 16-group conv & 2 & 1024$\\times$1$\\times$ 1 \\\\\\hline\n\\end{tabular}\n\\caption{The tianzige-CNN structure in Glyce.}\n\\label{cnn_structure}\n\\end{table}\n\\end{comment}\nDirectly using deep CNNs  \\citet{he2016deep,Szegedy_2016_CVPR,ma2018shufflenet} in our task results in very poor performances because of (1) relatively smaller size of the character images:\nthe size of Imagenet images is usually  at the scale of\n800*600, while the size of Chinese character images is significantly smaller, usually at the scale of 12*12;\n and (2) the lack of training examples:\n classifications on the imageNet dataset utilizes tens of millions of different images. In contrast,\nthere are only about 10,000 distinct Chinese characters. \nTo tackle these issues,\nwe propose the Tianzige-CNN structure, which is tailored to Chinese character modeling\nas illustrated in Figures \\ref{TIANZIGE}. \nTianzige (\u7530\u5b57\u683c) is a traditional form of Chinese Calligraphy. It is a four-squared format (similar to Chinese character \u7530) for beginner to learn writing Chinese characters. \nThe input image $x_{\\text{image}}$ \nis first passed through a convolution layer with kernel size 5 and output channels 1024 to capture lower level graphic features.\nThen a max-pooling layer of kernel size 4 is applied to the feature map and reduces the resolution from $8\\times 8$ to the tianzige $2\\times 2$, .  \nThis $2\\times 2$ tianzige structure presents how radicals are arranged in Chinese characters and also the order by which Chinese characters are written. \nWe find the tianzige structure of significant importance in extracting character meanings. \nFinally, we apply convolutional operations to map tianzige grids to the final outputs.\nInstead of using conventional convolutional operations, we use group convolutions \\citep{krizhevsky2012imagenet,zhang2017interleaved}.\nGroup convolutional filters are much smaller than their normal counterparts, and thus are less prune to overfitting. \nIt is fairly easy to adjust the model from  single script to multiple scripts, which can be achieved by simply changing the input from 2D (i.e., $d_{\\text{font}}\\times d_{\\text{font}}$) to 3D (i.e., $N_{\\text{script}}\\times d_{\\text{font}}\\times d_{\\text{font}}$), where $d_{\\text{font}}$ denotes the font size.\n\n\\subsection{Image Classification as an Auxiliary Objective}\nTo further prevent overfitting, we use the task of image classification as an auxiliary training objective. \nThe glyph embedding $h_{\\text{image}}$ from CNNs\nwill be forwarded to an image classification objective to predict its corresponding charID.\nSuppose the label of image $x$ is $z$.\nThe training objective  for the image classification task $\\mathcal{L}(\\text{cls})$ is given as follows: \n\\begin{equation}\n\\begin{aligned}\n\\mathcal{L}(\\text{cls}) & = -\\log p(z|x)\\\\\n& = -\\log \\text{softmax} (W\\times h_{\\text{image}})\n\\end{aligned}\n\\end{equation}\nLet  $\\mathcal{L}(\\text{task})$ denote the task-specific objective for the task we need to tackle, e.g., language modeling, word segmentation, Machine Comprehension, etc. \nWe linearly combine $\\mathcal{L}(\\text{task})$ and $\\mathcal{L}(\\text{cl})$, making the final objective training function as follows:\n\\begin{equation}\n\\mathcal{L} =   (1-\\lambda(t)) \\ \\mathcal{L}(\\text{task}) + \\lambda(t) \\mathcal{L}(\\text{cls})\n\\end{equation}\nwhere $\\lambda(t)$ controls the trade-off between the task-specific objective and the auxiliary image-classification objective. \n$\\lambda$ is a function of the number of epochs $t$:\n$\\lambda(t) = \\lambda_0 \\lambda_1^{t}$, where $\\lambda_0\\in [0,1]$ denotes the starting value,   $\\lambda_1\\in [0,1]$ denotes the decaying value. \nThis means that the influence from the image classification objective \n decreases as the training proceeds, with the intuitive explanation being that at the earlier stage of training, we need more regulations from the image classification task. \nAdding image classification as a training objective mimics the idea of multi-task learning.\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=5.5in]{newfigs\/overview_combine.png}\n\\centering\n\\caption{Combing glyph information with BERT.}\n\\label{overview_combine}\n\\end{figure*}\n\n\\subsection{ Combing Glyph Information with BERT}\nThe glyph embeddings can be directly output to downstream models such as RNNs, LSTMs, transformers. \n\nSince large scale pretraining systems using language models, such as BERT \\citep{devlin2018bert}, ELMO \\citep{peters2018deep} and GPT \\citep{radford2018improving}, \n have proved to be effective in a wide range of NLP tasks,\n we explore the possibility of combining glyph embeddings with BERT embeddings.\n Such a strategy \n  will potentially endow the model with  the advantage of  both glyph evidence and large-scale pretraining. \n  The overview of the combination is shown in Figure \\ref{overview_combine}.\nThe  model consists of four  layers: the BERT layer, the glyph layer, the Glyce-BERT layer and the task-specific output layer. \n\n\\paragraph{BERT Layer}\nEach input sentence $S$ \nis concatenated with a special \\textsc{cls} token denoting the start of the sentence, and a \\textsc{sep} token, denoting the end of the sentence. \nGiven a pre-trained BERT model, the embedding for each token of $S$ is computed using BERT. \nWe use to output from the last layer of the BERT transformer to represent the current token. \n\n\\paragraph{Glyph Layer} the output from tianzege-CNNs.\n\n\\paragraph{Glyce-BERT layer}\nPosition embeddings are first added to the glyph embeddings. The addition is then concatenated with BERT to obtain the full Glyce representation. This marks the end of Glyph and BERT combination. \n\\paragraph{Task-specific output layer}\nGlyce-BERT representations are used to represent the token at that position, similar as word embeddings or Elmo emebddings \\citep{peters2018deep}.\nContextual-aware information has already been encoded in the BERT representation but not glyph representations. We thus need additional context   models to encode contextual-aware glyph representations. \nHere, we choose multi-layer transformers \\citep{vaswani2017attention}.  \nThe output representations from transformers are used as outputs to the prediction layer.  \nIt is worth noting that the representations for the special \\textsc{cls} and \\textsc{sep} positions are maintained at the final task-specific embedding layer. \n\\begin{figure*}\n\\center\n\\includegraphics[width=5.5in]{newfigs\/cls.png}\n\\centering\n\\caption{Using Glyce-BERT model for different tasks. }\n\\label{cls}\n\\end{figure*}\n\n\n\\section{Tasks}\nIn this section, we describe how glyph embeddings can be used for different NLP tasks. \nIn the vanilla version, glyph embeddings are simply treated as character embeddings, which are fed to \n models built on top of the word-embedding layers, such as RNNs, CNNs or more sophisticated ones. \nIf combined with BERT, we need to specifically handle the integration between the glyph embeddings and the pretrained embeddings from BERT in different scenarios, as will be discussed in order below:\n\n\\subsection{Sequence Labeling Tasks}\nMany Chinese NLP tasks, such as name entity recognition (NER), Chinese word segmentation (CWS) and part speech tagging (POS), can be formalized as character-level sequence\nlabeling tasks, in which  we need to predict a label for each character. \nFor glyce-BERT model, \nthe embedding output from the task-specific layer (described in Section 2,4)\n is fed to the CRF model for label predictions. \n\\subsection{Single Sentence Classification}\nFor text classification tasks, a single label is to be predicted for the entire sentence. \nIn the  BERT model, the representation for the \\textsc{cls} token in the final layer of BERT is output to the softmax layer for prediction. \nWe adopt the similar strategy, in which the representation for the \\textsc{cls} token in the task-specific layer is fed to the softmax layer to predict labels. \n\\subsection{Sentence Pair Classification} \nFor sentence pair classification task like SNIS \\citep{DBLP:conf\/emnlp\/BowmanAPM15}, a model needs to handle the interaction between the two sentences and outputs a label for a pair of sentences. In the  BERT setting, a sentence pair $(s_1, s_2)$ is concatenated with one \\textsc{cls} and two \\textsc{sep} tokens, denoted by \n[\\textsc{cls}, $s_1$, \\textsc{sep}, $s_2$, \\textsc{sep}]. The concatenation is fed to the BERT model, and the \nobtained\n\\textsc{cls} representation is then fed to the softmax layer for label prediction. \nWe adopt the similar strategy for Glyce-BERT, in which [\\textsc{cls}, $s_1$, \\textsc{sep}, $s_2$, \\textsc{sep}] is subsequently passed through the BERT layer, Glyph layer, Glyce-BERT layer and the task-specific output layer. The \\textsc{cls} representation from the task-specific output layer is fed to the softmax function for the final label prediction. \n\n\n\\section{Experiments}\nTo enable apples-to-apples comparison, we perform grid  parameter search for both baselines and the proposed model on the dev set. \nTasks that we work on are described in order below.\n\\subsection{Tagging}\n\\paragraph{NER}\nFor the task of Chinese NER, we used the widely-used OntoNotes, MSRA, Weibo and resume datasets.\nSince most datasets don't have gold-standard segmentation, the task is normally treated as \n a char-level \n  tagging task: outputting an NER tag for each character. \nThe currently most widely used non-BERT model is Lattice-LSTMs \\citep{DBLP:journals\/corr\/abs-1810-12594, DBLP:conf\/acl\/ZhangY18}, \n achieving better performances than CRF+LSTM \\citep{DBLP:conf\/acl\/MaH16}. \n \\paragraph{CWS}:\nThe task of Chinese word segmentation (CWS) is \nnormally treated\nas a char-level tagging problem. \nWe used the widely-used PKU, MSR, CITYU and AS benchmarks from SIGHAN 2005 bake-off for evaluation.\n\n\\paragraph{POS}\nThe task of Chinese part of speech tagging is normally formalized as a character-level sequence labeling task, assigning labels to each of the character within the sequence. \nWe use the CTB5, CTB9 and   UD1 (Universal Dependencies) benchmarks to test our models. \n\n\\begin{table}[!ht]\n \\small\n\\caption{Results for NER tasks.}\n\\label{NER}\n\\centering\n\\begin{multicols}{2}\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{OntoNotes}\\\\\\hline\nModel & P & R & F \\\\\\hline\nCRF-LSTM & 74.36 & 69.43&71.81 \\\\\nLattice-LSTM  & 76.35& 71.56& 73.88  \\\\\nGlyce+Lattice-LSTM&82.06& 68.74&74.81  \\\\\n&&&{\\bf (+ 0.93) }\\\\\\hline\nBERT& 78.01& 80.35& 79.16 \\\\\nGlyce+BERT & 81.87 & 81.40 & 80.62 \\\\\n&&&{\\bf (+1.46)}\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{MSRA}\\\\\\hline\nModel & P & R & F \\\\\\hline\nCRF-LSTM&92.97&90.80&91.87\\\\\nLattice-LSTM  & 93.57& 92.79& 93.18  \\\\\nLattice-LSTM+Glyce& 93.86&93.92 & 93.89\\\\\n&&&{\\bf (+0.71)}\\\\\\hline\nBERT& 94.97 & 94.62 & 94.80 \\\\\nGlyce+BERT & 95.57 & 95.51 & 95.54 \\\\\n&&&{\\bf(+0.74)} \\\\\\hline\n\\end{tabular}\n\n\\columnbreak\n\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{Weibo}\\\\\\hline\nModel & P & R & F \\\\\\hline\nCRF-LSTM&51.16&51.07&50.95\\\\\nLattice-LSTM&52.71&53.92&53.13 \\\\\nLattice-LSTM+Glyce&53.69&55.30& 54.32\\\\\n&&&{\\bf (+1.19)} \\\\\\hline\nBERT& 67.12 & 66.88 & 67.33 \\\\\nGlyce+BERT & 67.68& 67.71& 67.60 \\\\\n&&&{\\bf (+0.27)}\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{resume}\\\\\\hline\nModel & P & R & F \\\\\\hline\nCRF-LSTM&94.53 & 94.29&94.41 \\\\\nLattice-LSTM  & 94.81& 94.11&  94.46  \\\\\nLattice-LSTM+Glyce&95.72&95.63& 95.67 \\\\\n&&&{\\bf (+1.21)} \\\\\\hline\nBERT& 96.12 & 95.45 & 95.78 \\\\\nGlyce+BERT & 96.62 & 96.48 & 96.54 \\\\\n&&&{\\bf(+0.76)}\\\\\\hline\n\\end{tabular}\n\\end{multicols}\n\\end{table}\n\nResults for NER, CWS and POS are respectively shown in Tables \\ref{NER}, \\ref{CWS} and \\ref{POS}. \nWhen comparing with non-BERT models, Lattice-Glyce performs better than all non-BERT models\n across all datasets in all tasks. \nBERT outperforms non-BERT models in all datasets except Weibo. \n This is due to the discrepancy  between the dataset which BERT is pretrained on (i.e., wikipedia) and weibo. \nThe Glyce-BERT model outperforms BERT and sets new SOTA results across all datasets, manifesting the effectiveness of incorporating glyph information. \nWe are able to achieve SOTA performances on all of the datasets using either Glyce model itself or BERT-Glyce model. \n\n\\begin{table}[!ht]\n\\center\n\\small\n\\caption{Results for CWS tasks.}\n\\label{CWS}\n\\centering\n\\begin{multicols}{2}\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{PKU}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{DBLP:conf\/acl\/YangZD17}&-&- &96.3\\\\\n\\citet{DBLP:journals\/corr\/abs-1808-06511}&-&- & 96.1\\\\\n\\citet{DBLP:journals\/corr\/abs-1903-04190}&-&- &96.6\\\\\nBERT&96.8 & 96.3 &96.5 \\\\\nGlyce+BERT&97.1&96.4&96.7 \\\\\n&&&{\\bf(+0.2)}\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{MSR}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{DBLP:conf\/acl\/YangZD17}&-&-&97.5\\\\\n\\citet{DBLP:journals\/corr\/abs-1808-06511}&-&- & 98.1\\\\\n\\citet{DBLP:journals\/corr\/abs-1903-04190}&-&-&97.9\\\\\nBERT& 98.1 & 98.2 & 98.1\\\\\nGlyce+BERT&98.2 & 98.3 &98.3 \\\\\n&&&{\\bf(+0.2)}\\\\\\hline\\hline\n\\end{tabular}\n\n\\columnbreak\n\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{CITYU}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{DBLP:conf\/acl\/YangZD17}&-&- &96.9\\\\\n\\citet{DBLP:journals\/corr\/abs-1808-06511}&-&- & 97.2\\\\\n\\citet{DBLP:journals\/corr\/abs-1903-04190}&-&- &97.6\\\\\nBERT& 97.5 & 97.7 & 97.6\\\\\nGlyce+BERT &97.9 &98.0 &97.9\\\\\n&&&{\\bf(+0.3)}\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{AS}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{DBLP:conf\/acl\/YangZD17}&-&-&95.7\\\\\n\\citet{DBLP:journals\/corr\/abs-1808-06511}&-&- & 96.2\\\\\n\\citet{DBLP:journals\/corr\/abs-1903-04190}&-&- &96.6\\\\\nBERT& 96.7 &96.4 &96.5 \\\\\nGlyce+BERT& 96.6 &96.8 & 96.7 \\\\\n&&&{\\bf(+0.2)}\\\\\\hline\\hline\n\\end{tabular}\n\\end{multicols}\n\\end{table}\n\n\n\n\n\n\\begin{table}[!ht]\n\\small\n\\caption{Results for POS tasks.}\n\\label{POS}\n\\begin{multicols}{2}\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{CTB5}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{shao2017character} (Sig)&93.68&94.47&94.07\\\\\n\\citet{shao2017character} (Ens)&93.95&94.81&94.38\\\\\nLattice-LSTM  & 94.77& 95.51& 95.14 \\\\\nGlyce+Lattice-LSTM&95.49&95.72& 95.61\\\\\n&&&{\\bf (+0.47)} \\\\\\hline\nBERT& 95.86 & 96.26 & 96.06 \\\\\nGlyce+BERT &  96.50 & 96.74 & 96.61 \\\\\n&&& {\\bf(+0.55) }\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{CTB6}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{shao2017character} (Sig)&-&-&90.81\\\\\nLattice-LSTM  & 92.00& 90.86&91.43  \\\\\nGlyce+Lattice-LSTM& 92.72& 91.14&  91.92\\\\\n&&&{\\bf (+0.49)} \\\\\\hline\nBERT& 94.91 & 94.63 & 94.77 \\\\\nGlyce+BERT &  95.56 & 95.26 & 95.41\\\\\n&&&{\\bf(+0.64)}\\\\\\hline\\hline\n\\end{tabular}\n\n\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{CTB9}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{shao2017character} (Sig)&91.81&94.47&91.89\\\\\n\\citet{shao2017character} (Ens)&92.28&92.40&92.34\\\\\nLattice-LSTM  & 92.53& 91.73& 92.13 \\\\\nLattice-LSTM+Glyce&92.28 &92.85 &92.38 \\\\\n&&&{\\bf (+0.25)} \\\\\\hline\nBERT& 92.43 & 92.15 & 92.29 \\\\\nGlyce+BERT & 93.49 & 92.84 & 93.15 \\\\\n&&&{\\bf(+0.86)}\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{UD1}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{shao2017character} (Sig)&89.28&89.54&89.41\\\\\n\\citet{shao2017character} (Ens)&89.67&89.86&89.75\\\\\nLattice-LSTM&90.47&89.70& 90.09\\\\\nLattice-LSTM+Glyce&91.57 & 90.19&90.87 \\\\\n&&&{\\bf (+0.78)} \\\\\\hline\nBERT& 95.42 & 94.17 & 94.79 \\\\\nGlyce+BERT & 96.19 & 96.10 & 96.14 \\\\\n&&&{\\bf (+1.35) }\\\\\\hline\\hline\n\\end{tabular}\n\\end{multicols}\n\\end{table}\n\n\n\\subsection{Sentence Pair Classification}\nFor sentence pair classification tasks, we need to output a label for each pair of sentence.\nWe employ the following four different datasets: (1) {\\bf BQ} (binary classification task) \\citep{DBLP:conf\/emnlp\/BowmanAPM15}; \n(2) {\\bf LCQMC} (binary classification task)  \\citep{liu2018lcqmc},\n(3) {\\bf XNLI} (three-class classification task) \\citep{williamsmulti}, and\n(4) {\\bf NLPCC-DBQA}  (binary classification task) \\footnote{\\url{https:\/\/github.com\/xxx0624\/QA_Model}}.\n\n\n\nThe current  non-BERT SOTA model is based on the  bilateral\nmulti-perspective matching model (BiMPM) \\citep{DBLP:conf\/ijcai\/WangHF17}, which specifically tackles the subunit matching between sentences. \nGlyph embeddings are incorporated into BiMPMs, forming the Glyce+BiMPM baseline. \nResults regarding each model on different datasets are given in  Table \\ref{pair}. \nAs can be seen, BiPMP+Glyce outperforms BiPMPs, achieving the best results among non-bert models. \nBERT outperforms all non-BERT models, and\nBERT+Glyce performs the best, setting new SOTA results on all of the four benchmarks. \n\n\n\\begin{table}[!ht]\n\\small\n\\caption{Results for sentence-pair classification tasks.}\n\\label{pair}\n\\begin{multicols}{2}\n\\begin{tabular}{|lllll|}\\hline\n\\multicolumn{5}{|c|}{BQ}\\\\\\hline\nModel & P & R & F & A \\\\\\hline\nBiMPM & 82.3 & 81.2 & 81.7 & 81.9  \\\\\nGlyce+BiMPM & 81.9 & 85.5 & 83.7 & 83.3  \\\\\n&&&{\\bf (+2.0)}&{\\bf (+1.4)}\\\\\\hline\nBERT & 83.5 & 85.7 & 84.6 & 84.8\\\\\nGlyce+BERT & 84.2 & 86.9 & 85.5 & 85.8\\\\\n&&&{\\bf (+0.9)}&{\\bf(+1.0)}\\\\\\hline\\hline\n\\multicolumn{5}{|c|}{XNLI}\\\\\\hline\nModel & P & R & F & A \\\\\\hline\nBiMPM & - & - & - &67.5 \\\\\nGlyce+BiMPM & - & - & - & 67.7 \\\\\n&&&&{\\bf(+0.2)}\\\\\\hline\nBERT & - & - & - & 78.4\\\\\nGlyce+BERT & - & - & - & 79.2\\\\\n&&&&{\\bf(+0.8)}\\\\\\hline\n\\end{tabular}\n\n\\begin{tabular}{|lllll|}\\hline\n\\multicolumn{5}{|c|}{LCQMC}\\\\\\hline\nModel & P & R & F & A \\\\\\hline\nBiMPM & 77.6 & 93.9 & 85.0 & 83.4  \\\\\nGlyce+BiMPM & 80.4 & 93.4 & 86.4 & 85.3  \\\\\n&&&{\\bf (+1.4)}&{\\bf (+1.9)} \\\\\\hline\nBERT & 83.2 & 94.2 & 88.2 &87.5 \\\\\nGlyce+BERT & 86.8 & 91.2 & 88.8 & 88.7\\\\\n&&&{\\bf(+0.6)}&{\\bf(+1.2)}\\\\\\hline\\hline\n\\multicolumn{5}{|c|}{NLPCC-DBQA}\\\\\\hline\nModel & P & R & F & A \\\\\\hline\nBiMPM &78.8&56.5&65.8&-\\\\\nGlyce+BiMPM & 76.3& 59.9&67.1&-  \\\\\n&&&{\\bf(+1.3)}&-\\\\\\hline\nBERT&  79.6&86.0&82.7&-\\\\\nGlyce+BERT &81.1&85.8&83.4&-\\\\\n&&&{\\bf(+0.7)}&-\\\\\\hline\n\\end{tabular}\n\\end{multicols}\n\\end{table}\n\n\n\\begin{table*}[!ht]\n\\small\n\\center\n\\caption{Accuracies for Single Sentence Classification task. }\n\\label{single}\n\\begin{tabular}{|cccc|}\\hline\nModel & ChnSentiCorp & the Fudan corpus & iFeng \\\\\\hline\n LSTM & 91.7& 95.8&84.9\\\\\nLSTM + Glyce & 93.1 & 96.3 & 85.8  \\\\\n& {\\bf (+ 1.4) } & {\\bf (+0.5)} & {\\bf (+0.9)} \\\\\\hline\nBERT & 95.4 & 99.5 & 87.1   \\\\\nGlyce+BERT & 95.9 & 99.8 & 87.5   \\\\\n&{\\bf(+0.5)}&{\\bf(+0.3)}&{\\bf(+0.4)}\\\\\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Single Sentence Classification}\nFor single sentence\/document  classification, we need to output a label for a text sequence. The label could be either a sentiment indicator or a news genre. Datasets that we use include: (1) ChnSentiCorp (binary classification); (2) the Fudan corpus (5-class classification) \\citep{fudan}; and \n(3) Ifeng (5-class classification). \n\n\nResults for different models on different tasks are shown in Table \\ref{single}. \nWe observe similar phenomenon as before:  \nGlyce+BERT achieves SOTA results on all of the datasets. \nSpecifically, the Glyce+BERT model achieves an almost perfect accuracy (99.8) on the the Fudan corpus. \n\n\\subsection{Dependency Parsing and Semantic Role Labeling}\nFor dependency parsing \\citep{chen2014fast,dyer2015transition}, we used the\nwidely-used\n Chinese Penn Treebank 5.1 dataset for evaluation. Our implementation uses the previous state-of-the-art Deep Biaffine \n model \\citet{dozat2016deep} as a backbone. \nWe\n replaced the word vectors from the biaffine model with Glyce-word embeddings, and exactly followed its model structure and  training\/dev\/test split criteria. \n We report scores for\nunlabeled attachment score (UAS)\nand\nlabeled attachment score (LAS).\nResults\nfor previous models\n are copied from \\citep{dozat2016deep,ballesteros2016training,cheng2016bi}.\nGlyce-word pushes  SOTA performances up by +0.9 and +0.8 in terms of UAS and LAS scores.\n\n\\begin{table}\n\\small\n\\begin{tabular}{|lcc|}\\hline\n\\multicolumn{3}{|c|}{Dependency Parsing}\\\\\\hline\nModel & UAS & LAS \\\\\\hline\n\\citet{ballesteros2016training}& 87.7 &86.2\\\\\n\\citet{kiperwasser2016simple}&87.6& 86.1\\\\\n\\citet{cheng2016bi}& 88.1 &85.7\\\\\n Biaffine & 89.3 & 88.2 \\\\\\hline\n \\multirow{ 2}{*}{Biaffine+Glyce}&  90.2&89.0\\\\\n & {\\bf (+0.9)}&{\\bf (+0.8)}\\\\\\hline\n\\end{tabular}\n\\begin{tabular}{|cccc|}\\hline\n\\multicolumn{4}{|c|}{Semantic Role Labeling}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{roth2016neural}& 76.9 &73.8 &75.3\\\\\n\\citet{marcheggiani2017encoding}&84.6 &80.4 &82.5\\\\ \n\\citet{he2018syntax} &      84.2 & 81.5&  82.8 \\\\\\hline\n \\multirow{ 2}{*}{k-order pruning+Glyce}& 85.4&82.1 & 83.7  \\\\\n &{\\bf (+0.8)}&{\\bf (+0.6)}&{\\bf (+0.9)}\\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nFor \nthe task of\nsemantic role labeling (SRL) \\citep{roth2016neural,marcheggiani2017encoding,he2018syntax}, we used the CoNLL-2009 shared-task. \nWe used the current SOTA model, the k-order \npruning algorithm \\citep{he2018syntax} as a backbone.\\footnote{Code open sourced at \\url{https:\/\/github.com\/bcmi220\/srl_syn_pruning}}\nWe replaced word embeddings with Glyce embeddings.\nGlyce outperforms the previous SOTA performance by  0.9 with respect to the F1 score, achieving a new SOTA score of  83.7.\n\nBERT does not perform competitively in these two tasks, and results are thus omitted. \n\n\n\\begin{comment}\n\\section{Ablation Studies}\nIn this section, we discuss the influence of different factors of the proposed model. \nWe use the LCQMC dataset of the sentence-pair prediction task for illustration. \nFactors that we discuss include training strategy,  model architecture, auxiliary image-classification objective, etc. \n\\subsection{Training Strategy}\nSection \\ref{training-tactic} talks about a training tactic (denoted by BERT-first-glyce-second-joint), in which given task-specific supervisions,  we first fine-tune the BERT model, then freeze BERT to fine-tune the glyph layer, and finally jointly tune both layers until convergence. \nWe compare this strategy with other tactics, including (1)\nthe {\\it Glyph-first-Joint} strategy, in which BERT is not fine-tuned in the beginning: we first freeze BERT to tune the glyph layer, and then jointly tune \nboth layers until convergence; and (2) the\n{\\it joint-first} strategy, in which we directly jointly training two models until converge. \n\\begin{table}[!ht]\n\\center\n\\caption{Impact of different training strategy.}\n\\label{training-tactic}\n\\begin{tabular}{|lllll|}\\hline\nStrategy&P & R & F & A\\\\\\hline\nBiMPM & 77.6 & 93.9 & 85.0 & 83.4  \\\\\nGlyce+BiMPM & 80.4 & 93.4 & 86.4 & 85.3  \\\\\nonly BERT  & 83.2 & 94.2 & 88.2 & 87.5\\\\\\hline\nGlyph-first-Joint & 82.5 & 94.0 & 87.9 & 87.1 (-0.4) \\\\\njoint-first  & 81.5 & 95.1 & 87.8 & 86.8 (-0.7) \\\\\nBERT-first-glyce-second-joint & 86.8 & 91.2 & 88.8 & 88.7  (+1.2)\\\\\\hline\n\\end{tabular}\n\\end{table}\n\nResults are shown in Table \\ref{training-tactic}. \nIt shows the significant importance of the order by which different layers are trained. \nThe Glyph-first-Joint and joint-first models not only underperform the proposed tactic, but also the vanilla version of the BERT model. \nThe is due to the fact that\nthe BERT layer is pretrained but the glyph layer is randomly initialized. \nGiven the relatively small amount of training signals, the BERT layer could be mislead by the randomly initialized glyph layer at the early stage of training, leading to inferior final performances.  \n\n\n\n\\subsection{Structures of the task-specific output layer}\nThe concatenation of the glyph embedding and the BERT embedding is fed to the task-specific output layer. The task-specific output layer is made up with two layers of transformers. Here we change transformers to other structures such as BiLSTMs and CNNs\n to explore the influence.\nWe also try the  BiMPM structure \\citep{DBLP:conf\/ijcai\/WangHF17} to see the result. \n\n\\begin{table}[!ht]\n\\center\n\\caption{Impact of structures for the task-specific output layer.}\n\\label{structure}\n\\begin{tabular}{lllll}\\hline\nStrategy &P & R & F & A\\\\\\hline\nTransformers & 86.8 & 91.2 & 88.8 & 88.7 \\\\\nBiLSMTs & 81.8 & 94.9 & 87.9 & 86.9\\\\\nCNNs & 81.5 & 94.8 & 87.6 & 86.6\\\\\nBiMPM & 81.1 & 94.6 & 87.3 & 86.2\\\\\\hline\n\\end{tabular}\n\\end{table}\n\nPerformances are shown in Table \\ref{structure}. \nAs can be seen, transformers not only outperform BiLSTMs and CNNs, but also the BiMPM  structure, which is specifically built for the sentence pair classification task. \nWe conjecture that this is because of the consistency between transformers and BERT. \n\\subsection{The image-classification training objective}\nWe also explore the influence of the  image-classification training objective, which outputs the glyph representation to an image-classification objective. \nTable \\ref{image} represents its influence. As can be seen, this auxiliary training objective given a +0.8 performance boost. \n\n\\begin{table}[!ht]\n\\center\n\\caption{Impact of the auxilliary image-classification training objective.}\n\\label{image}\n\\begin{tabular}{lllll}\\hline\nStrategy&P & R & F & A\\\\\\hline\nW image-cls  & 86.8 & 91.2 & 88.8 & 88.7\\\\\nWO image-cls & 83.9 & 93.6 & 88.4 & 87.9\\\\\\hline\n\\end{tabular}\n\\end{table}\n\n\\end{comment}\n\n\\section{Ablation Studies}\nIn this section, we discuss the influence of different factors of the proposed model. \nWe use the LCQMC dataset of the sentence-pair prediction task for illustration. \nFactors that we discuss include training strategy,  model architecture, auxiliary image-classification objective, etc. \n\\subsection{Training Strategy}\nThis section talks about a training tactic (denoted by BERT-glyce-joint), in which given task-specific supervisions,  we first fine-tune the BERT model, then freeze BERT to fine-tune the glyph layer, and finally jointly tune both layers until convergence. \nWe compare this strategy with other tactics, including (1)\nthe {\\it Glyph-Joint} strategy, in which BERT is not fine-tuned in the beginning: we first freeze BERT to tune the glyph layer, and then jointly tune \nboth layers until convergence; and (2) the\n{\\it joint} strategy, in which we directly jointly training two models until converge. \n\\begin{table}[!ht]\n\\center\n\\begin{tabular}{lllll}\\hline\nStrategy&Precision & Recall & F1 & Accuracy\\\\\\hline\nBERT-glyce-joint & 86.8 & 91.2 & 88.8 & 88.7 \\\\\nGlyph-Joint & 82.5 & 94.0 & 87.9 & 87.1\\\\\njoint  & 81.5 & 95.1 & 87.8 & 86.8 \\\\\nonly BERT  & 83.2 & 94.2 & 88.2 & 87.5\\\\\\hline\n\\end{tabular}\n\\caption{Impact of different training strategy.}\n\\label{training-tactic}\n\\end{table}\n\nResults are shown in Table \\ref{training-tactic}. As can be seen, the BERT-glyce-joint outperforms the rest two strategies. \nOur explanation for the inferior performance of the {\\it joint} strategy is as follows: \nthe BERT layer is pretrained but the glyph layer is randomly initialized. \nGiven the relatively small amount of training signals, the BERT layer could be mislead by the randomly initialized glyph layer at the early stage of training, leading to inferior final performances.  \n\n\\subsection{The image-classification training objective}\nWe also explore the influence of the  image-classification training objective, which outputs the glyph representation to an image-classification objective. \nTable \\ref{image} represents its influence. As can be seen, this auxiliary training objective given a {+0.8} performance boost. \n\\begin{table}[!ht]\n\\center\n\\begin{tabular}{lllll}\\hline\nStrategy&Precision & Recall & F1 & Accuracy\\\\\\hline\nW image-cls  & 86.8 & 91.2 & 88.8 & 88.7\\\\\nWO image-cls & 83.9 & 93.6 & 88.4 & 87.9\\\\\\hline\n\\end{tabular}\n\\caption{Impact of the auxilliary image-classification training objective.}\n\\label{image}\n\\end{table}\n\n\\subsection{Structures of the task-specific output layer}\nThe concatenation of the glyph embedding and the BERT embedding is fed to the task-specific output layer. The task-specific output layer is made up with two layers of transformers. Here we change transformers to other structures such as BiLSTMs and CNNs\n to explore the influence.\nWe also try the  BiMPM structure \\cite{DBLP:conf\/ijcai\/WangHF17} to see the result. \n\\begin{table}[!ht]\n\\center\n\\begin{tabular}{lllll}\\hline\nStrategy &Precision & Recall & F1 & Accuracy\\\\\\hline\nTransformers & 86.8 & 91.2 & 88.8 & 88.7 \\\\\nBiLSMTs & 81.8 & 94.9 & 87.9 & 86.9\\\\\nCNNs & 81.5 & 94.8 & 87.6 & 86.6\\\\\nBiMPM & 81.1 & 94.6 & 87.3 & 86.2\\\\\\hline\n\\end{tabular}\n\\caption{Impact of structures for the task-specific output layer.}\n\\label{structure}\n\\end{table}\n\nPerformances are shown in Table \\ref{structure}. \nAs can be seen, transformers not only outperform BiLSTMs and CNNs, but also the BiMPM  structure, which is specifically built for the sentence pair classification task. \nWe conjecture that this is because of the consistency between transformers and the BERT structure. \n\n\\subsection{CNN structures} \nResults for different CNN structures are shown  in Table \\ref{CNN}. As can be seen, the adoption of tianzige-CNN structure introduces a performance boost of F1 about +1.0. Directly using deep CNNs in our task results in very poor performances because of (1) relatively smaller size of the character images: the size of ImageNet images is usually  at the scale of 800*600, while the size of Chinese character images is significantly smaller, usually at the scale of 12*12; and (2) the lack of training examples: classifications on the ImageNet dataset utilizes tens of millions of different images. In contrast, there are only about 10,000 distinct Chinese characters. We utilize the Tianzige-CNN  (\u7530\u5b57\u683c) structures tailored to logographic character modeling for Chinese. This tianzige structure is of significant importance in extracting character meanings. \n\n\\begin{table}[!ht]\n\\small\n\\centering\n\\begin{tabular}{llll}\\hline\n &Precision & Recall & F1 \\\\\\hline\nVanilla-CNN & 85.3 & 89.8 & 87.4 \\\\\n\\citet{he2016deep}& 84.5&90.8& 87.5  \\\\\nTianzige-CNN & 86.8 & 91.2 & 88.8  \\\\\\hline\n\\end{tabular}\n\\caption{Impact of CNN structures.}\n\\label{CNN}\n\\end{table}\n\n\\section{Conclusion}\nIn this paper, we propose Glyce, Glyph-vectors for Chinese Character Representations. Glyce treats Chinese characters as images and uses Tianzige-CNN to extract character semantics.\nGlyce provides a general way to model character semantics of logographic languages. It is general and fundamental. Just like word embeddings,  Glyce can be integrated to any existing deep learning system. \n\\end{CJK*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supplementary Information}\n\n\n\\subsection{Theory}\n\nIn Eq.\\ (4) of the main text, we give the result of $\\langle \\mathbf{\\Omega}_\\mathrm{SO} \\rangle  =  \\int_0^{2\\pi} d\\theta \\, \\mathbf{\\Omega}_\\mathrm{SO} (\\mathbf{k} + \\mathbf{k}_\\mathrm{dr}) \\, f(\\theta)$ for the special case that drift occurs along the $y$-direction ($k_{\\mathrm{dr},x} = 0$) and detection at $x = 0$ ($k_{\\mathrm{di},x} = 0$). \nHere, we provide the result for a general case:\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO} \\rangle  = \\langle \\mathbf{\\Omega}_\\mathrm{SO,1} \\rangle  + \\langle \\mathbf{\\Omega}_\\mathrm{SO,3} \\rangle \\, ,\n\\end{equation}\n\nwith a term proportional to $\\beta_1$:\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO,1} \\rangle  =  \\frac{2 \\beta_1}{\\hbar} \n\\begin{pmatrix} \nk_{\\mathrm{di},y} + k_{\\mathrm{dr},y} \\\\\nk_{\\mathrm{di},x} + k_{\\mathrm{dr},x}\n\\end{pmatrix} \\, ,\n\\end{equation}\n\nand one proportional to $\\beta_3$:\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO,3} \\rangle  =  \\frac{2 \\beta_3}{\\hbar} \n\\begin{pmatrix} \n- k_{\\mathrm{di},y} -2 k_{\\mathrm{dr},y} + \\frac{2}{k_\\mathrm{F}^2} \\left[  k_{\\mathrm{di},y} k_{\\mathrm{dr},x}^2 + 2 k_{\\mathrm{di},x} k_{\\mathrm{dr},x}  k_{\\mathrm{dr},y} +  k_{\\mathrm{dr},x}^2 k_{\\mathrm{dr},y} - 3 k_{\\mathrm{di},y} k_{\\mathrm{dr},y}^2 -  k_{\\mathrm{dr},y}^3 \\right] \\\\\n- k_{\\mathrm{di},x} -2 k_{\\mathrm{dr},x} + \\frac{2}{k_\\mathrm{F}^2} \\left[  k_{\\mathrm{di},x} k_{\\mathrm{dr},y}^2 + 2 k_{\\mathrm{di},y} k_{\\mathrm{dr},y}  k_{\\mathrm{dr},x} +  k_{\\mathrm{dr},y}^2 k_{\\mathrm{dr},x} - 3 k_{\\mathrm{di},x} k_{\\mathrm{dr},x}^2 -  k_{\\mathrm{dr},x}^3 \\right]\n\\end{pmatrix}  \\, .\n\\end{equation}\n\nFor simplicity, we assumed $\\alpha = 0$ in the above expressions. \nWe now move to the special case where the electric field is applied along the $y$ direction, such that $k_{\\mathrm{dr},x} = 0$, and obtain\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO,3} \\rangle  = - \\frac{2 \\beta_3}{\\hbar} \n\\begin{pmatrix} \nk_{\\mathrm{di},y} \\left[1 + 6\\left(\\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}}\\right)^2 \\right] + 2 k_{\\mathrm{dr},y} \\left[1 + \\left(\\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}}\\right)^2 \\right] \\\\\nk_{\\mathrm{di},x} \\left[ 1 - \\left(\\frac{2  k_{\\mathrm{dr},y}}{k_\\mathrm{F}} \\right)^2 \\right]\n\\end{pmatrix}  \\, .\n\\end{equation}\n\n\nTo describe a measurement where spins are tracked along the drift direction $y$ and at $x = 0$, we additionally set $k_{\\mathrm{di},x} = 0$ and obtain\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO} \\rangle  =   \\frac{2 \\beta_1}{\\hbar} \n\\begin{pmatrix} \n k_{\\mathrm{di},y} + k_{\\mathrm{dr},y} \\\\\n0\n\\end{pmatrix} \n- \\frac{2 \\beta_3}{\\hbar} \n\\begin{pmatrix} \n k_{\\mathrm{di},y} \\left[1 + 6\\left(\\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}}\\right)^2 \\right] + 2 k_{\\mathrm{dr},y} \\left[1 + \\left(\\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}}\\right)^2 \\right]  \\\\\n0\n\\end{pmatrix}  \\, .\n\\end{equation}\n\nTranslating into a $(y, t)$ coordinate system via $y = \\frac{\\hbar}{m^*}t (k_{\\mathrm{dr},y} + k_{\\mathrm{di},y})$, we obtain\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO} \\rangle   t = q y + \\omega t \\, ,\n\\end{equation}\n\n\\noindent with\n\n\\begin{equation}\nq = \\frac{2 m^*}{\\hbar^2} \\left(  \\beta_1 - \\beta_3 \\left[  1 + 6 \\left(\\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}}\\right)^2 \\right] \\right)\n\\end{equation}\n\n\\noindent and\n\n\\begin{equation}\n\\omega = -\\frac{2 m^*}{\\hbar^2} v_\\mathrm{dr} \\beta_3 \\left[ 1 - 4 \\left( \\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}} \\right)^2  \\right] \\, .\n\\end{equation}\n\nThe higher-order terms are not observed in the experiment and therefore not discussed in the main text. However, they might be observable for larger drift vectors or smaller electron sheet densities.\n\n\n\n\n\n\\subsection{Simulations}\n\n\n\\begin{figure}\n\\includegraphics{SI_fig1.pdf}\n\\caption{Validity of the model. (a) to (c) Simulation data of $S_z(y,t)$ for $\\alpha = 0$~eVm, $\\alpha = 1.5 \\times 10^{-13}$~eVm and $\\alpha = 3 \\times 10^{-13}$~eVm, respectively. In all cases, $\\beta^* = 3 \\times 10^{-13}$~eVm, $\\beta_3 = 0.7 \\times 10^{-13}$~eVm and $v_\\mathrm{dr} = 38$~km\/s (violet dashed line). We find good agreement between the simulation and the model (green solid lines) for the entire parameter range.\n\\label{SI_fig1} }\n\\end{figure}\n\nIn the main text, we discuss the validity of the model for cases away from the PSH symmetry, i.e., away from $\\alpha = \\beta^*$, by comparing the model with spin-precession maps obtained from numerical Monte-Carlo simulations. We state that, as long as drift occurs along $y$, we obtain good agreement between simulation and model. \nIn Fig.\\ \\ref{SI_fig1}, we show the corresponding simulations for three different cases between $\\alpha \/\\beta^* = 0$ (isotropic) and $\\alpha \/\\beta^* = 1$ (PSH). The model of Eqs.\\ (5) and (6) of the main text (green solid lines) correctly predicts the simulated spin dynamics for the entire parameter range for drift along $y$.\n\n\n\n\n\\subsection{Fitting}\n\nEquation (1) in the main text contains six independent fit parameters. Suitable starting values for the fitting are obtained in the following way. For the amplitude $A_0$ we choose the value of $S_z (t=0, y=0)$. The drift velocity, $v_\\mathrm{dr}$, is defined by the shift of the spin packet in time and its starting value is estimated manually. The spin diffusion constant, $D_\\mathrm{s}$, is determined by the broadening of the Gaussian envelope function and we start with a typical value for samples from the same wafer. For the dephasing time, $\\tau$, we use 1~ns as a starting value. The most important parameters for the presented study are $\\omega$, the temporal precession frequency, and $q$, the spatial wavenumber. Both quantities are little affected by the other fit parameters. Starting values for both of them are obtained from a line-cut through the data at a fixed time (a fixed position) for $q$ (for $\\omega$). Before calculating the mean-squared error between Eq. (1) and the measured $S_z (y, t)$, we perform a one-dimensional convolution of Eq. (1) with the Gaussian intensity profiles of the pump and probe laser spots along $y$. This step is very important, because its neglect distorts particularly the value of $\\omega$. \n\n\nAll fit parameters are then constrained to a reasonable range. To determine each parameter's fit value and confidence interval, we vary that parameter in small steps through its full range. At each step, all other parameters are optimized to minimize the mean-squared error between the data and Eq. (1) by a Nelder-Mead simplex search algorithm. The value of the parameter with the smallest error defines the fit value. For all fit parameters, we find a single minimum. The confidence interval, as shown in Fig. 2 in the main text, is then defined by an increase of the mean-squared error by 5~\\% from its minimal value. The mean-squared error is taken over approximately 3000 data points (typically 35 steps of $t$, 85 steps of $y$ or $x$). \n\n\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:I}\nThe behavior of charged quantum particles inside a uniform magnetic field in \\linebreak \\mbox{Minkowski spacetime} displays quantized energy levels ---\\,the so-called Landau levels\\,--- that were first derived by Rabi based on the relativistic Dirac equation \\cite{Rabi} and by Landau based on the Schr\\\"odinger equation \\cite{Landau} (see e.g., Ref.\\,\\cite{LandauLifshitz} for a textbook introduction). This discovery led to many physical applications in condensed matter physics, such as the explanation of the Landau diamagnetism \\cite{Landau}, of the Shubnikov\u2013de Haas effect \\cite{Hadju}, of the de Haas-van Alphen effect \\cite{Shoenberg} and of the integer quantum Hall effect \\cite{QHEBook}. Remarkably, the Landau levels have even found astrophysical applications. In fact, the physics of neutron stars and other highly magnetized stellar objects is expected to be governed to some extent by the discrete nature of the energy levels occupied by the charged fermions moving under the influence of the intense magnetic field of such astrophysical objects \\cite{BookNeutronStar,WhiteDwarfs,WhiteDwarfs3,Mass-Radius,WhiteDwarfs4,Broderick,Chamel1,Chamel2,Yakovlev, NeutronStarMatter}. When we recall that such highly compact astrophysical objects are also sources of very intense gravitational fields, it becomes extremely important to investigate the fate of Landau levels under the influence of a combination of a magnetic field and the gravitational field of a spherical object. Such a study has recently been conducted in Refs.\\,\\cite{GravityLandauI,GravityLandauII,QHGravity} based on the Klein-Gordon and Schr\\\"odinger equations by considering the gravitational interaction of the particle with a massive sphere as a small perturbation compared to the interaction of the particle with the magnetic field. Although relying on the Schr\\\"odinger equation and treating gravity perturbatively as done in these studies might be sufficient for laboratory \\mbox{applications \\cite{Landry1,Landry2,COWBall,Josephson},} when seeking astrophysical applications, implementing a general-relativistic and a non-perturbative approach and appealing to the relativistic Dirac equation are highly recommended.  \n \nThe Dirac equation in a curved spacetime caused {\\it purely} by an intense magnetic field--known as the Melvin universe \\cite{Melvin}---has been solved exactly in Ref.\\,\\cite{Nunes} where the quantized energy levels were extracted. Contemplating curved spacetimes caused purely by a magnetic field is motivated by the well-known fact that the strong magnetic fields of neutron stars could reach up to  {$10^{12-15}$}\\,G  \n on the surface of magnetars \\cite{Magnetars}, and up to $10^{17}\\,$G in their interior (see, e.g., Refs.\\,\\cite{InsideNS1,InsideNS2} and the references therein). On the other hand, neutron stars are made of neutrons, as well as of a plasma of electrons, protons and muons, the energies of which are affected by such strong magnetic fields. The aim of the present paper is therefore to include the general-relativistic effect of the intense gravitational field of such stellar objects by studying the dynamics of charged fermions moving {\\it inside} a massive spherical object in the presence of a uniform magnetic field. To the best of our knowledge, such a study has not appeared in the literature before. Unlike the purely magnetic spacetime considered in Ref.\\,\\cite{Nunes}, however, we consider in the present work the curved spacetime caused purely by the distribution of matter inside the core of the stellar objects rather than by the uniform magnetic field. The contribution of the latter to spacetime curvature will thus be considered negligible. The justification behind such an assumption will be discussed in Section \\ref{sec:II}. \n\nThe remainder of this paper is structured as follows. In Section \\ref{sec:II}, we derive the Dirac equation in a general static and spherically symmetric curved spacetime in the presence of a uniform magnetic field. We discuss the various symmetries of the equation and the complications one faces when attempting to solve the equation exactly. In \\mbox{Section \\ref{sec:III}}, we apply the equation obtained in Section \\ref{sec:II} to the case of the interior Schwarzschild solution. We then extract the quantized energy levels of charged fermions moving along the equator inside a massive spherical object by keeping only the leading terms of the differential equation. In Section \\ref{sec:IV}, we briefly discuss the ways to overcome the limitations we encountered in solving our equation in Section \\ref{sec:III}. In Section \\ref{sec:V}, we apply our results to the study of the magnetization of neutron stars. We conclude this paper with a short discussion section in which we summarize our findings. \n\n\n\n\\section{Curved-Spacetime Dirac Equation in a Uniform Magnetic Field}\\label{sec:II}\nSince our aim in this paper is to consider the Dirac equation in the static and spherically symmetric interior Schwarzschild solution, let us start by deriving in this section the equation for a general static and spherically symmetric curved spacetime in the presence of a uniform magnetic field. For that purpose, let us write our spacetime metric in the following general form:\n\\begin{equation}\\label{GeneralMetric}\n    {\\rm d}s^2=-\\mathcal{A}^2(r)\\,c^2{\\rm d}t^2+\\mathcal{B}^2(r)\\,{\\rm d}r^2+r^2({\\rm d}\\theta^2+\\sin^2\\theta{\\rm d}\\phi^2),\n\\end{equation}\nwhere the radial functions $\\mathcal{A}(r)$ and $\\mathcal{B}(r)$ are everywhere regular except, maybe, at the origin $r=0$.  In addition, let us assume the constant and uniform magnetic field $\\bf B$\nto be parallel to the $z$-direction. A convenient gauge for the vector potential would then be $A_{\\mu}=(0,0,0,\\frac{1}{2}{Br^2\\sin^2\\theta})$. The Dirac equation of a particle of mass $m$ and of charge $e$ (that we take to be the negative charge of the electron for definiteness) moving inside a curved spacetime and minimally coupled to a Maxwell field $A_\\mu$ reads \\cite{Pollock,BookDiracInCS},\n\\begin{equation}\\label{GeneralDirac}\ni\\hbar e^\\mu_a\\gamma^a\\left(\\partial_\\mu+\\frac{1}{8}\\omega_\\mu^{\\;bc}\\,[\\gamma_b,\\gamma_c]-\\frac{ieA_\\mu}{\\hbar }\\right)\\psi=mc\\psi.\n\\end{equation}\nThe quantities $e^\\mu_a$ are the inverse of the spacetime tetrads $e^a_\\mu$, whereas $\\gamma^a$ are the gamma matrices, $\\omega_\\mu^{\\,bc}$ is the spin connection and $[\\gamma_b,\\gamma_c]$ is the commutator of the gamma matrices. The spin connection is antisymmetric, $\\omega_\\mu^{\\;bc}=-\\omega_\\mu^{\\;cb}$, and it is expressed in terms of the tetrad fields and the Christoffel symbols $\\Gamma_{\\mu\\nu}^\\lambda$ of the spacetime by $\\omega_\\mu^{\\;bc}=e^b_\\nu\\partial_\\mu e^{c\\nu}+\\Gamma_{\\mu\\nu}^\\lambda e^b_\\lambda e^{c\\nu}$. \n\n\n\\newpage\nOn the other hand, two choices for the four curved-spacetime gamma matrices $\\gamma^\\mu=e^a_\\mu\\gamma^a$ are possible. One may choose either to work with the tetrads' axes parallel to the $r$, $\\theta$ and $\\phi$ coordinates, or choose to orient the tetrads' axes parallel to some rectangular coordinate system \\cite{BrillWheeler}. When combined with the standard representation of the constant flat-spacetime gamma matrices $\\gamma^a$, given explicitly by\n\\begin{equation}\\label{DiracMatrices}\n\\gamma^0=\n\\begin{pmatrix}\n\\mathbbm{1} & 0\\\\\n0 & -\\mathbbm{1}\n\\end{pmatrix},\\qquad\n\\gamma^k=\n\\begin{pmatrix}\n0 & \\sigma_k\\\\\n-\\sigma_k & 0\n\\end{pmatrix},\n\\end{equation}\nwhere $\\mathbbm{1}$ is the $2\\times2$ unit matrix and $\\sigma_k$ (for $k=1,2,3$) are the three Pauli matrices, the curved-space gamma matrices $\\gamma^\\mu$ take on a simpler form in the first case than in the second \\cite{BrillWheeler}. The spinor wavefunction $\\psi$ in the two cases will be different, but physical quantities as well as the radial wave equation will all be the same as the two representations are simply related by a unitary transformation \\cite{BrillWheeler}. However, to deal with the mixed cylindrical and spherical symmetries imposed, respectively, by the magnetic and gravitational fields, it is easier to adopt the spherical-coordinates representation of the gamma matrices, in which the three Pauli matrices take the form\n\\begin{equation}\\label{PauliM}\n\\sigma_r=\n\\begin{pmatrix}\n\\cos\\theta & \\sin\\theta e^{-i\\phi}\\\\\n\\sin\\theta e^{i\\phi} & -\\cos\\theta\n\\end{pmatrix},\\qquad\n\\sigma_\\theta=\n\\begin{pmatrix}\n-\\sin\\theta & \\cos\\theta e^{-i\\phi}\\\\\n\\cos\\theta e^{i\\phi} & \\sin\\theta\n\\end{pmatrix},\\qquad \n\\sigma_\\phi=\n\\begin{pmatrix}\n0 & -ie^{-i\\phi}\\\\\nie^{i\\phi} & 0\n\\end{pmatrix}.\n\\end{equation}\nWe now choose the spacetime tetrads for the metric (\\ref{GeneralMetric}) to be $e^a_\\mu={\\rm diag}(\\mathcal{A},\\mathcal{B},r,r\\sin\\theta)$. With these tetrads, we easily evaluate the nonvanishing components of the contracted spin connection with the commutator of the gamma matrices to be:\n\\begin{align}\n    \\tfrac{1}{8}\\omega_0^{ab}[\\gamma_a,\\gamma_b]&=\\frac{\\mathcal{A}'}{2\\mathcal{B}}\\gamma^0\\gamma^1,\\qquad \\tfrac{1}{8}\\omega_2^{ab}[\\gamma_a,\\gamma_b]=\\frac{1}{2\\mathcal{B}}\\gamma^1\\gamma^2,\\nonumber\\\\ \\tfrac{1}{8}\\omega_3^{ab}[\\gamma_a,\\gamma_b]&=\\frac{\\sin\\theta}{2\\mathcal{B}}\\gamma^1\\gamma^3+\\frac{\\cos\\theta}{2}\\gamma^2\\gamma^3.\n\\end{align}\nPlugging these, together with the vector potential $A_\\mu$, into Eq.\\,(\\ref{GeneralDirac}) and then multiplying the whole equation on the left by $\\beta=\\gamma^0$ and using the definition $\\alpha^k=\\beta\\gamma^k$ of the Dirac alpha matrices, the resulting equation reads,\n\\begin{equation}\\label{ExplicitGeneralDirac}\n      \\left[\\frac{i\\partial_t}{\\mathcal{A}c}\\psi+\\frac{i\\alpha^1}{\\mathcal{B}}\\left(\\partial_r+\\frac{1}{r}+\\frac{\\mathcal{A}'}{2\\mathcal{A}}\\right)+\\frac{i\\alpha^2}{r}\\left(\\partial_\\theta+\\frac{\\cot\\theta}{2}\\right)+\\frac{i\\alpha^3}{r\\sin\\theta}\\left(\\partial_\\phi-\\frac{ieA_\\phi}{\\hbar}\\right)\\right]\\psi=\\frac{mc}{\\hbar}\\beta\\psi.\n\\end{equation}\nWe are interested in this paper in stationary states of energy $E$, for which we have $\\partial_t\\psi=-\\tfrac{i}{\\hbar}E\\psi$. On the other hand, we may simplify further Eq.\\,(\\ref{ExplicitGeneralDirac}) by setting $\\psi=\\Psi\/(r\\sqrt{\\mathcal{A}\\sin\\theta})$. With these two replacements, the resulting equation in $\\Psi$ takes the form\n\\begin{equation}\\label{SimplifiedGeneralDirac}\n    \\Bigg[\\frac{E}{\\mathcal{A}\\hbar c}+\\frac{i\\alpha^1}{\\mathcal{B}}\\partial_r+\\frac{i\\alpha^2}{r}\\partial_\\theta+\\frac{i\\alpha^3}{r\\sin\\theta}\\left(\\partial_\\phi-\\frac{ieB}{2\\hbar }r^2\\sin^2\\theta\\right)-\\frac{mc}{\\hbar}\\beta\\Bigg]\\Psi=0.\\\\\n\\end{equation}\n\nThis is the general equation obeyed by fermions under the influence of the uniform magnetic field. Unlike the well-known case of the Dirac equation in a spherically symmetric potential, i.e., in a central potential (see, e.g., Ref.\\,\\cite{Greiner}), our Eq.\\,(\\ref{SimplifiedGeneralDirac}) may not be separated in the radial variables $r$ and the angular variables $\\theta$ and $\\phi$. This is due to the mixture of the spherical symmetry imposed by the gravitational field and the cylindrical symmetry imposed by the vertical uniform magnetic field $\\bf B$.\nOne might then be tempted to switch to cylindrical coordinates $(\\rho,\\phi,z)$ to benefit from the cylindrical symmetry pertaining to the magnetic field. Unfortunately, even doing so would not make Eq.\\,(\\ref{SimplifiedGeneralDirac}) easier to solve. In fact, by making the substitutions $\\rho=r\\sin\\theta$ and $z=r\\cos\\theta$, Eq.\\,(\\ref{SimplifiedGeneralDirac}) would contain operators of the form $(\\rho^2+z^2)^{-\\frac{1}{2}}z\\,\\partial_\\rho$ and $(\\rho^2+z^2)^{-\\frac{1}{2}}\\rho\\,\\partial_z$. It is then evident that even if we neglect the $z$-variation of $\\Psi$ by assuming $\\partial_z\\Psi\\approx0$, i.e., by neglecting the vertical momentum of the particle, we would still be left with an equation that is not separable in the variables $\\rho$, $z$ and $\\phi$. \n\nIt seems then that the only choice left is to set $z=0$ as well. Physically, this is due (in the {\\it absence} of any other interaction of the particle) to the unstable nature of any position $z\\neq0$, caused by the transverse gravitational attraction that pulls the particle away from any plane other than the equator. This observation can also be reached by \nreferring to the variable $\\theta$ in Eq.\\,(\\ref{SimplifiedGeneralDirac}). In fact, the only way of removing the $\\theta$-dependence in that equation is to set $\\theta=\\frac{\\pi}{2}$, i.e., to restrict ourselves to particles moving along the equatorial plane. This is because setting $\\theta=\\theta_0\\neq\n\\frac{\\pi}{2}$ would prevent us from letting $\\rho=r\\sin\\theta_0$ vary and have at the same time planar motion. If $\\theta_0$ is set to a constant other than $\\frac{\\pi}{2}$, then for $\\rho$ to change, $r$ must change too, which makes the particle leave the plane $\\theta=\\theta_0$. Only along the equatorial plane $\\theta_0=\\frac{\\pi}{2}$ does $\\rho=r\\sin\\frac{\\pi}{2}=r$ become a variable describing \\mbox{planar motion}. \n\nWe will come back to the possibility of considering a plane other than the equator in Section \\ref{sec:IV}. Before setting $\\theta=\\frac{\\pi}{2}$, however, we should first extract the general single differential equation from the coupled Dirac spinors.   \n\n\n\\section{The Quantized Energy Levels}\\label{sec:III}\nTo extract a quantization condition, if any, on the energy $E$ from Eq.\\,(\\ref{SimplifiedGeneralDirac}), we need first to decompose the Dirac spinor into a pair of $2$-spinors as follows\n\\begin{equation}\n    \\Psi(r,\\theta,\\phi)=\n    \\begin{bmatrix}\n    \\Phi(r,\\theta,\\phi)\\\\\n    \\Theta(r,\\theta,\\phi)\n\\end{bmatrix}.\n\\end{equation}\nSubstituting this ansatz into Eq.\\,(\\ref{SimplifiedGeneralDirac}), the latter splits into the following coupled first-order differential equations:\n\\begin{align}\\label{SplitEqsSpinors}\n\\left(\\frac{E}{\\mathcal{A}\\hbar c}-\\frac{mc}{\\hbar}\\right)\\Phi+\\left[\\frac{i\\sigma_r}{\\mathcal{B}}\\partial_r+\\frac{i\\sigma_\\theta}{r}\\partial_\\theta+\\frac{i\\sigma_\\phi}{r\\sin\\theta}\\left(\\partial_\\phi-\\frac{ieB}{2\\hbar }r^2\\sin^2\\theta\\right)\\right]\\Theta=0,\\nonumber\\\\\n\\left(\\frac{E}{\\mathcal{A}\\hbar c}+\\frac{mc}{\\hbar}\\right)\\Theta+\\left[\\frac{i\\sigma_r}{\\mathcal{B}}\\partial_r+\\frac{i\\sigma_\\theta}{r}\\partial_\\theta+\\frac{i\\sigma_\\phi}{r\\sin\\theta}\\left(\\partial_\\phi-\\frac{ieB}{2\\hbar }r^2\\sin^2\\theta\\right)\\right]\\Phi=0.\n\\end{align}\nIt is clear from these coupled differential equations that given the $\\rho$-dependence of all the coupling terms, extracting one of the 2-spinors from one equation and substituting it into the other would result in a very complicated single equation. This is unlike what happens with the standard Dirac equation for the case of the central potential of the hydrogen atom \\cite{Greiner}. It would be natural then at this stage to attempt to solve this system of equations by applying the approach developed in Refs.\\,\\cite{Alhaidari1,Alhaidari2,Alhaidari3}, which consists of applying a gauge transformation followed by a unitary transformation to turn one of the coupling terms in one of the two equations into a constant. Indeed, such a method considerably simplifies the resulting final single equation one arrives at. Unfortunately, such an approach cannot be consistently applied in our case because that approach is specifically designed only for a certain type of interactions of the particle with the gauge field, which is not the case for our present physical system\\footnote{FH is grateful to Prof. A.\\,D. Alhaidari for the helpful discussion on this subtle point.}. The other possibility would then be to attempt to rather adopt the approach developed in Ref.\\,\\cite{Alhaidari4} for the Dirac equation in curved spacetime. Unfortunately, that approach is not helpful to us either as it becomes rapidly involved even when the particle is not coupled to a gauge potential. For this reason, such an approach has been applied in Ref.\\,\\cite{Alhaidari4} only in $1+1$ dimensions. We are thus left only with the old strategy for dealing with coupled differential equations of the type (\\ref{SplitEqsSpinors}).\n\nExtracting the 2-spinor $\\Theta$ from the second line of Eq.\\,(\\ref{SplitEqsSpinors}) and plugging it into the first, we obtain the following single equation for the 2-spinor $\\Phi$:\n\\begin{align}\\label{2-SpinorEq}\n&\\frac{\\Phi_{,rr}}{\\mathcal{B}^2}+\\left(\\frac{2}{\\mathcal{B}r}-\\frac{\\mathcal{B}_{,r}}{\\mathcal{B}^3}+\\frac{\\mathcal{A}_{,r}E}{\\mathcal{A}\\mathcal{B}^2\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi_{,r}\\nonumber\\\\\n&+\\frac{\\Phi_{,\\theta\\theta}}{r^2}+\\left(\\frac{i\\left[\\mathcal{B}-1\\right]\\sigma_\\phi}{\\mathcal{B}r^2}+\\frac{\\cot\\theta}{r^2}+\\frac{i\\sigma_\\phi}{r}\\frac{\\mathcal{A}_{,r}E}{\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi_{,\\theta}\\nonumber\\\\\n&+\\frac{\\Phi_{,\\phi\\phi}}{r^2\\sin^2\\theta}\n+\\left(\\frac{i\\left[1-\\mathcal{B}\\right]\\sigma_\\theta}{\\mathcal{B}r^2\\sin\\theta}-\\frac{ieB}{\\hbar}-\\frac{i\\sigma_\\theta}{r\\sin\\theta}\\frac{\\mathcal{A}_{,r}E}{\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi_{,\\phi}\\nonumber\\\\\n&+\\Bigg(\\frac{E^2}{\\mathcal{A}^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}-\\frac{e^2B^2r^2\\sin^2\\theta}{4\\hbar^2}-\\frac{\\left[\\mathcal{B}+1\\right]eB\\sigma_\\theta\\sin\\theta}{2\\hbar\\mathcal{B}}+\\frac{\\sigma_reB\\cos\\theta}{\\hbar}\\nonumber\\\\\n&\\qquad-\\frac{\\mathcal{A}_{,r}EeBr\\,\\sigma_\\theta\\sin\\theta}{2\\hbar\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\Bigg)\\Phi=0.\n\\end{align}\nWith this result, we can now proceed to restrict the motion of the particle to the equatorial plane by setting $\\theta=\\frac{\\pi}{2}$. Therefore, we also need to perform the replacement $r\\sin\\theta=r\\equiv\\rho$ everywhere in this equation. On the other hand, for the $\\theta$-derivatives, we use the following identities valid for motion restricted to a plane: $\\partial_\\theta=r\\cos\\theta\\partial_\\rho$ and $\\partial_\\theta^2=-r\\sin\\theta\\partial_\\rho+r^2\\cos^2\\theta\\partial_\\rho^2$. Therefore, for $\\theta=\\frac{\\pi}{2}$ we should discard the partial derivative $\\partial_\\theta$ and perform the replacement  $\\partial_\\theta^2\\rightarrow-\\rho\\,\\partial_\\rho$. The above equation then becomes\n\\begin{align}\\label{Plane2-SpinorEq}\n&\\frac{\\Phi_{,\\rho\\rho}}{\\mathcal{B}^2}+\\left(\\frac{2-\\mathcal{B}}{\\mathcal{B}\\rho}-\\frac{\\mathcal{B}_{,\\rho}}{\\mathcal{B}^3}+\\frac{\\mathcal{A}_{,\\rho}E}{\\mathcal{A}\\mathcal{B}^2\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi_{,\\rho}\\nonumber\\\\\n&+\\frac{\\Phi_{,\\phi\\phi}}{\\rho^2}\n+\\left(\\frac{i\\left[\\mathcal{B}-1\\right]\\sigma_z}{\\mathcal{B}\\rho^2}-\\frac{ieB}{\\hbar}+\\frac{i\\sigma_z}{\\rho}\\frac{\\mathcal{A}_{,\\rho}E}{\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi_{,\\phi}\\nonumber\\\\\n&+\\left(\\frac{E^2}{\\mathcal{A}^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}-\\frac{e^2B^2\\rho^2}{4\\hbar^2}+\\frac{\\left[\\mathcal{B}+1\\right]eB\\sigma_z}{2\\hbar\\mathcal{B}}+\\frac{\\mathcal{A}_{,\\rho}EeB\\rho\\,\\sigma_z}{2\\hbar\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi=0.\n\\end{align}\nWe have used here the fact that according to Eq.\\,(\\ref{PauliM}), the spherical-coordinates Pauli matrix $\\sigma_\\theta$ reduces to $-\\sigma_z$ for $\\theta=\\frac{\\pi}{2}$ (as expected), where $\\sigma_z$ is the usual third Pauli matrix in Cartesian coordinates. \n\nOn the other hand, according to expressions (\\ref{PauliM}) of the Pauli matrices, we see in Eq.\\,(\\ref{Plane2-SpinorEq}) that all the terms are diagonal and that the equation does not involve the angular variable $\\phi$ explicitly. Therefore, without loss of generality, we can write the two independent solutions $\\Phi_+(\\rho,\\phi)$ and $\\Phi_-(\\rho,\\phi)$ of the equation in a basis made of the spin-eigenstates along the $z$-direction with eigenvalues $s=\\pm1$, respectively. We thus introduce arbitrary radial functions $f_+(\\rho)$ and $f_-(\\rho)$ such that\n\\begin{equation}\\label{Sigma3Basis}\n\\Phi_+(\\rho)=e^{i\\ell\\phi}\n\\begin{bmatrix}\nf_+(\\rho)\\\\0\n\\end{bmatrix},\\qquad \n\\Phi_-(\\rho)=e^{i\\ell\\phi}\n\\begin{bmatrix}\n0\\\\f_-(\\rho)\n\\end{bmatrix}.\n\\end{equation}\nImposing the periodic condition along the equator, $\\Phi(\\rho,\\phi+2\\pi)=\\Phi(\\rho,\\phi)$, we learn that the quantum number $\\ell$ must be an integer number, which we take here to be nonnegative in accordance with the negative sign we chose for the charge $e$. Plugging these ansatzes into Eq.\\,(\\ref{Plane2-SpinorEq}) yields the following second-order differential equation in $\\rho$:\n\\begin{align}\\label{fEq}\n&\\frac{f_s''}{\\mathcal{B}^2}+\\left(\\frac{2-\\mathcal{B}}{\\mathcal{B}\\rho}-\\frac{\\mathcal{B}'}{\\mathcal{B}^3}+\\frac{\\mathcal{A}'E}{\\mathcal{A}\\mathcal{B}^2\\left[E+\\mathcal{A}mc^2\\right]}\\right)f'_s\\nonumber\\\\\n&+\\Bigg(\\frac{E^2}{\\mathcal{A}^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}+\\frac{eB\\ell}{\\hbar}-\\frac{e^2B^2\\rho^2}{4\\hbar^2}-\\frac{\\mathcal{B}\\ell^2+\\left[\\mathcal{B}-1\\right]s\\ell}{\\mathcal{B}\\rho^2}+\\frac{\\left[\\mathcal{B}+1\\right]eBs}{2\\hbar\\mathcal{B}}\\nonumber\\\\\n&\\qquad-\\frac{\\mathcal{A}'Es\\left[2\\hbar\\ell-eB\\rho^2\\right]}{2\\hbar\\rho\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\Bigg)f_s=0,\n\\end{align}\nHere, and henceforth, a prime denotes a derivative with respect to the variable $\\rho$, and the function $f_s(\\rho)$ stands for the two cases $f_\\pm(\\rho)$ of $s=\\pm1$, respectively. Next, we introduce the following ansatz: \\begin{equation}\\label{FirstAnsatz}\n    f_s(\\rho)=\\left(\\frac{\\mathcal{B}}{\\mathcal{A}}E+\\mathcal{B}mc^2\\right)^{\\frac{1}{2}}F_s(\\rho),\n\\end{equation}\nfor some radial function $F_s(\\rho)$. Please note that only the convergence of the radial function $F_s(\\rho)$ is required henceforth, for the argument of the square root in this ansatz converges for both large and small values of $\\rho$ as we shall see below after introducing the explicit components of the spacetime metric. Plugging now this ansatz into Eq.\\,(\\ref{fEq}), the latter takes the following form:\n\\begin{multline}\\label{SimplifiedfEq}\n\\!\\!\\!\\frac{F_s''}{\\mathcal{B}^2}+\\frac{2-\\mathcal{B}}{\\mathcal{B}\\rho}F'_s+\\bigg[\\frac{E^2}{\\mathcal{A}^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}+\\frac{eB\\ell}{\\hbar}-\\frac{e^2B^2\\rho^2}{4\\hbar^2}-\\frac{\\mathcal{B}\\ell^2+(\\mathcal{B}-1)s\\ell}{\\mathcal{B}\\rho^2}\\\\\n+\\frac{(\\mathcal{B}+1)eBs}{2\\hbar\\mathcal{B}}-\\frac{\\mathcal{A}'Es(2\\hbar\\ell-eB\\rho^2)}{2\\hbar\\rho\\mathcal{A}\\mathcal{B}(E+\\mathcal{A}mc^2)}-\\frac{1}{2\\mathcal{B}^2}\\left(\\frac{\\mathcal{A}'E}{\\mathcal{A}[E+\\mathcal{A}mc^2]}-\\frac{\\mathcal{B}'}{\\mathcal{B}}\\right)'\\\\\n-\\left(\\frac{\\mathcal{A}'E}{2\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{\\mathcal{B}'}{2\\mathcal{B}^2}\\right)^2-\\frac{2-\\mathcal{B}}{2\\mathcal{B}\\rho}\\left(\\frac{\\mathcal{A}'E}{\\mathcal{A}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{\\mathcal{B}'}{\\mathcal{B}}\\right)\\bigg]F_s=0.\n\\end{multline}\nFinally, the following ansatz,\n\\begin{equation}\\label{FinalAnsatz}\n    F_s(\\rho)=G_s(\\rho)\\exp\\left[\\int\\frac{(1-\\mathcal{B})^2}{2\\rho}{\\rm d}\\rho\\right],\n\\end{equation}\nfor an arbitrary radial function $G_s(\\rho)$, allows us to convert Eq.\\,(\\ref{SimplifiedfEq}) into the following more useful form:\n\\begin{align}\\label{FinalfEq}\n&G''_s+\\frac{G'_s}{\\rho}+\\bigg[\n\\frac{E^2}{\\mathcal{A}^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}+\\frac{eB\\ell}{\\hbar}-\\frac{e^2B^2\\rho^2}{4\\hbar^2}-\\frac{\\ell^2}{\\rho^2}+\\frac{(1-\\mathcal{B})s\\ell}{\\mathcal{B}\\rho^2}+\\frac{(1+\\mathcal{B})eBs}{2\\hbar\\mathcal{B}}\\nonumber\\\\\n&-\\frac{\\mathcal{A}'Es\\left[2\\hbar\\ell-eB\\rho^2\\right]}{2\\hbar\\rho\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{1}{2\\mathcal{B}^2}\\left(\\frac{\\mathcal{A}'E}{\\mathcal{A}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{\\mathcal{B}'}{\\mathcal{B}}\\right)'-\\left(\\frac{\\mathcal{A}'E}{2\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{\\mathcal{B}'}{2\\mathcal{B}^2}\\right)^2\\nonumber\\\\\n&-\\frac{2-\\mathcal{B}}{2\\mathcal{B}\\rho}\\left(\\frac{\\mathcal{A}'E}{\\mathcal{A}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{\\mathcal{B}'}{\\mathcal{B}}\\right)-\\frac{\\mathcal{B}'(1-\\mathcal{B})}{\\mathcal{B}^2\\rho}-\\frac{(1-\\mathcal{B})^4}{4\\mathcal{B}^2\\rho^2}\\bigg]\\mathcal{B}^2G_s=0.\n\\end{align}\n\nIt is clear that Eq.\\,(\\ref{FinalfEq}) would not be easy to solve exactly. However, we can clearly see now which terms of the equation could safely be dropped out even without making any prior assumption on the orders of magnitude of the various physical quantities involved, such as the energy of the particle and the strengths of the gravitational and magnetic interaction terms. In Section \\ref{sec:IV}, we shall come back to this point to discuss the order of magnitude of the correction that would have been brought to our final result for the energy levels of the particle if we had kept those extra terms.\n\nNow, while Eq.\\,(\\ref{FinalfEq}) seems analytically challenging to solve even when keeping only the dominant terms, the fact that the metric components $\\mathcal{A}^2$ and $\\mathcal{B}^2$ of the interior Schwarzschild solution depend on $\\rho^2$ instead of $\\rho$ will greatly simplify our task as we shall see shortly. For a spherical body of mass $M$, of radius $R$ and of uniform density, the interior Schwarzschild solution describing the gravitational field inside the body is given by the metric (\\ref{GeneralMetric}), with \\cite{Synge}\n\\begin{equation}\\label{InteriorSchwar}\n    \\mathcal{A}(\\rho)=\\frac{3}{2}\\left(1-\\frac{r_s}{R}\\right)^{\\frac{1}{2}}-\\frac{1}{2}\\left(1-\\frac{r_s\\rho^2}{R^3}\\right)^{\\frac{1}{2}},\\qquad \\mathcal{B}(\\rho)=\\left(1-\\frac{r_s\\rho^2}{R^3}\\right)^{-1\/2},\n\\end{equation}\nwhere $r_s=2GM\/c^2$ is the Schwarzschild radius of the massive body. For convenience, we introduce the dimensionless parameter $\\eta=\\frac{3}{2}(1-\\frac{r_s}{R})^{\\frac{1}{2}}$ and the inverse length squared $\\lambda=\\frac{r_s}{R^3}$. Please note that by choosing this metric, we have ignored the possible geometric effect of the magnetic field on the spacetime. We will come back to this point in Section \\ref{sec:IV} as well.   \n\nLet us now show the convergence of the square root in our ansatz (\\ref{FirstAnsatz}). Please note that for very small values of $\\rho$, the functions (\\ref{InteriorSchwar}) approximate to $\\mathcal{A}\\sim\\eta-\\frac{1}{2}+\\frac{\\lambda}{4}\\rho^2$ and $\\mathcal{B}\\sim1+\\frac{\\lambda}{2}\\rho^2$, from which we see that the leading-order $\\rho$-dependent term in the argument of the square root in Eq.\\,(\\ref{FirstAnsatz}) is proportional to $\\rho^2$. This establishes thus the convergence of that square root for very small $\\rho$. On the other hand, for $\\rho\\geq R$, the functions (\\ref{InteriorSchwar}) lead, by continuity, to the metric components of the {\\it exterior} Schwarzschild solution, for we have then $\\mathcal{A}=\\mathcal{B}^{-1}=(1-\\frac{r_s}{\\rho})^{\\frac{1}{2}}$. Therefore, for very large values of $\\rho$, the functions $\\mathcal{A}$ and $\\mathcal{B}$ in the argument of the square root in our ansatz (\\ref{FirstAnsatz}) approximate to $\\mathcal{A}\\sim1-\\frac{r_s}{2\\rho}$ and $\\mathcal{B}\\sim1+\\frac{r_s}{2\\rho}$. This implies that the leading-order $\\rho$-dependent term in the argument of the square root is proportional to $1\/\\rho$. This establishes thus the convergence of that square root for large values of $\\rho$ as well, whence we deduce the convergence of that square root for all values of $\\rho$. \n\nNext, we use the functions (\\ref{InteriorSchwar}) to expand $1\/\\mathcal{A}$ and $1\/\\mathcal{A}^2$, up to the first order in the parameter $\\lambda$, as follows:\n\\begin{align}\\label{ABPowerSeries}\n    \\frac{1}{\\mathcal{A}}&\\sim\\frac{2}{(2\\eta-1)}-\\frac{\\lambda\\rho^2}{(2\\eta-1)^2},\\nonumber\\\\\n    \\frac{1}{\\mathcal{A}^{2}}&\\sim\\frac{4}{(2\\eta-1)^2}-\\frac{4\\lambda\\rho^2}{(2\\eta-1)^3}.\n   \n   \n\\end{align}\nExpanding only up to the first order in $\\lambda$ is justified by the order of magnitude of $\\lambda$ (even for neutron stars as we will see in Section \\ref{sec:IV}). On the other hand, we also have $1\/\\mathcal{B}\\sim1-\\frac{\\lambda}{2}\\rho^2$ and $\\mathcal{B}^2\\sim1+\\lambda\\rho^2$. With these expansions, Eq.\\,(\\ref{SimplifiedfEq}) takes, up to the first order in the parameter $\\lambda$, the following form:\n\\begin{align}\\label{1stOrderFinalfEq}\n&\\frac{{\\rm d}^2G_s}{{\\rm d}\\rho^2}+\\frac{1}{\\rho}\\frac{{\\rm d}G_s}{{\\rm d}\\rho}+\\Bigg[\n\\frac{4E^2}{(2\\eta-1)^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}+\\frac{eB(\\ell+s)}{\\hbar}+\\frac{\\lambda eBs}{4\\hbar}-\\frac{\\lambda(2\\ell^2+s\\ell-2)}{2}\\nonumber\\\\\n&-\\frac{2\\lambda(s\\ell+1)}{(2\\eta-1)[2+(2\\eta-1)\\frac{mc^2}{E}]}-\\frac{\\ell^2}{\\rho^2}-\\Bigg(\\frac{e^2B^2}{4\\hbar^2}+\\frac{\\lambda m^2c^2}{\\hbar^2}+\\frac{8\\lambda E^2[\\eta-1]}{\\hbar^2c^2[2\\eta-1]^3}+\\frac{\\lambda eB[2\\ell+s]}{2\\hbar}\\nonumber\\\\\n&+\\frac{\\lambda eBs}{\\hbar(2\\eta-1)[2+(2\\eta-1)\\frac{mc^2}{E}]}\\Bigg)\\rho^2-\\frac{\\lambda e^2B^2}{4\\hbar^2}\\rho^4\\Bigg]G_s=0.\n\\end{align}\nThis equation has the form\n\\begin{equation}\\label{GEquation}\n    G_s''+\\frac{G_s'}{\\rho}+\\left(\\varepsilon-\\frac{\\ell^2}{\\rho^2}-\\kappa^2\\rho^2-\\tau^2\\rho^4\\right)G_s=0,\n\\end{equation}\nwhere a prime denotes a derivative with respect to $\\rho$ and the constants $\\varepsilon$, $\\kappa^2$ and $\\tau^2$ are straightforwardly read off from Eq.\\,(\\ref{1stOrderFinalfEq}). Setting $G_s=\\chi_s\/\\sqrt{\\rho}$ in Eq.\\,(\\ref{GEquation}), the latter takes the form\n\\begin{equation}\\label{Anharmonic}\n    \\chi''_s+\\left(\\varepsilon-\\frac{\\ell^2-\\frac{1}{4}}{\\rho^2}-\\kappa^2\\rho^2-\\tau^2\\rho^4\\right)\\chi_s=0.\n\\end{equation}\nThis is a Schr\\\"odinger equation with a centrifugal barrier term and an isotropic quartic anharmonic oscillator potential term. Please note that (i) in the case of a weak gravitational interaction compared to the magnetic one, i.e., for $\\lambda\\ll |e|B\/\\hbar$, and\/or (ii) in the case of charged fermions moving very close to the center of the spherical mass, i.e., for $\\rho\\ll R$, the purely quartic term $\\tau^2\\rho^4$ can be dropped out. In that case, Eq.\\,(\\ref{Anharmonic}) reduces to the one that has already been solved exactly in Ref.\\,\\cite{GravityLandauI}, where the quantized eigenvalues $\\varepsilon_{n,\\ell}$ have been extracted. We will come back to these special cases in Section \\ref{sec:IV-1}. Here, we will extract the eigenvalues from Eq.\\,(\\ref{Anharmonic}) by keeping the quartic term.\n\nIt is shown in Ref.\\,\\cite{JMP(1986)} that even though  Eq.\\,(\\ref{Anharmonic}) is not solvable exactly, one may still extract an approximate analytical expression for its eigenvalues using the Jeffreys\u2013Wentzel\u2013Kramers\u2013\\\\Brillouin (JWKB) approximation. To the fourth order in the approximation, the result reads \\cite{JMP(1986)}\n\\begin{equation}\\label{QuantizedEpsilon}\n\\varepsilon_{n,\\ell,s}=(2 \\tau)^{\\frac{2}{3}}\\sum_{k=0}^8\\left[\\frac{3\\sqrt{\\pi}\\,\\Gamma(\\tfrac{3}{4})}{\\sqrt{2}\\,\\Gamma(\\tfrac{1}{4})}\\left(2n+1+\\ell\\right)\\right]^{\\frac{4-2k}{3}}N_k\\left(\\kappa,\\tau,\\ell\\right),\n\\end{equation}\nwhere the explicit expressions of the nine terms $N_k(\\kappa, \\tau,\\ell)$ are given in Appendix \\ref{sec:App}. It is clear from this expression that the usual flat-space Landau levels of a charged fermion moving under the influence of a uniform magnetic field are dramatically altered. In fact, instead of the splitting of the levels that occurs due to a weak gravitational field \\mbox{outside \\cite{GravityLandauI}} or inside a spherical mass \\cite{QHGravity}, what happens here is a complete redistribution of the levels. Both the separation of the latter and their strengths are redefined for each principle quantum number $n$. \n\nFor the sake of illustration and simplicity, we display here explicitly the expression we obtain when keeping only the first term of the sum in Eq.\\,(\\ref{QuantizedEpsilon}). By substituting the constants $\\varepsilon$, $\\kappa$ and $\\tau$, as well as the term $N_0=1$ (as given in Appendix \\ref{sec:App}) into \\mbox{Eq.\\,(\\ref{QuantizedEpsilon}),} we find\n\\begin{align}\\label{QEpsilonN0}\n   E_{n,\\ell,s}&=\\left(\\eta-\\frac{1}{2}\\right)\\Bigg\\{m^2c^4-\\hbar c^2eB\\left(\\ell+s+\\frac{\\lambda s}{4}\\right)+\\frac{\\lambda\\hbar^2c^2(2\\ell^2+s\\ell-2)}{2}\\nonumber\\\\\n   &+\\frac{2\\lambda\\hbar^2c^2(s\\ell+1)}{(2\\eta-1)[2+(2\\eta-1)\\frac{mc^2}{E}]}+\\hbar^2c^2\\left[\\left(2n+1+\\ell\\right)\\frac{3\\,\\Gamma(\\tfrac{3}{4})}{\\Gamma(\\tfrac{1}{4})}\\right]^{\\frac{4}{3}}\\left(\\frac{\\pi eB\\sqrt{\\lambda}}{2\\hbar}\\right)^\\frac{2}{3}\\Bigg\\}^{\\frac{1}{2}}.\n\\end{align}\nThis does not, in fact, look at all like the familiar Landau levels. It must be noted here that this formula does not reduce to the flat-space result when setting $\\lambda=0$ because the expansion in terms of the JWKB integrals is about the pure quartic oscillator levels, not the other way around \\cite{JMP(1986)}. Therefore, to achieve a high accuracy in the Formula (\\ref{QuantizedEpsilon}), $\\kappa^2$ should not be larger than $\\tau^2$. To properly recover the flat-space case, one needs to set $\\lambda=0$ in Eq.\\,(\\ref{1stOrderFinalfEq}). Similarly, when the gravitational interaction is weaker than the magnetic interaction, one needs to extract the corresponding Landau levels by starting from Eq.\\,(\\ref{1stOrderFinalfEq}) as we shall do now.\n\\subsection{Weak Quartic Term}\\label{sec:IV-1}\nWhen the gravitational interaction is weak compared to the magnetic interaction and\/or when the charged particles move very closely to the center of the spherical mass, the quartic term $\\tau^2\\rho^4$ in Eq.\\,(\\ref{GEquation}) can safely be dropped out so that the equation takes the form,\n\\begin{equation}\\label{GEquationWithoutQuartic}\n    G_s''+\\frac{G_s'}{\\rho}+\\left(\\varepsilon-\\frac{\\ell^2}{\\rho^2}-\\kappa^2\\rho^2\\right)G_s=0.\n\\end{equation}\nThe solution to this equation is easily found to be given in terms of the confluent hypergeometric function as follows (see Ref.\\,\\cite{GravityLandauI} for the details of the derivation):\n\\begin{equation}\n    G_s(\\rho)=C\\rho^{\\ell}e^{-\\frac{|\\kappa|}{2}\\rho^2}\\,_1F_1\\left(\\frac{1+\\ell}{2}-\\frac{\\varepsilon}{4|\\kappa|};\\ell+1;|\\kappa|\\rho^2\\right).\n\\end{equation}\nThis solution is one of the two independent solutions of Eq.\\,(\\ref{1stOrderFinalfEq}) that is finite at the origin $\\rho=0$. $C$ is one of the constants of integration, and$\\,_1F_1(a;b;\\xi)$ is the confluent hypergeometric function of a variable $\\xi$ for any values of the parameters $a$ and $b$, except when $b$ is a negative integer for which case the function has a simple pole \\cite{HandBook}. Therefore, to guarantee a square-integrable character for the wavefunction, the following condition must be imposed for an arbitrary positive integer $n$:\n\\begin{equation}\n    \\frac{1+\\ell}{2}-\\frac{\\varepsilon}{4|\\kappa|}=-n,\n\\end{equation}\nfor which case, the confluent hypergeometric function becomes indeed a polynomial of finite degree $n$. By substituting into this condition, the values of $\\varepsilon$ and $\\kappa$ as read off from \\mbox{Eq.\\,(\\ref{1stOrderFinalfEq})}, we arrive at the following condition involving the energy $E$ of the \\mbox{charged fermion:}\n\\begin{align}\n&\\frac{4E^2}{(2\\eta-1)^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}+\\frac{eB(\\ell+s)}{\\hbar}+\\frac{\\lambda eBs}{4\\hbar}-\\frac{\\lambda(2\\ell^2+s\\ell-2)}{2}\\nonumber\\\\\n&-\\frac{2\\lambda(s\\ell+1)}{(2\\eta-1)[2+(2\\eta-1)\\frac{mc^2}{E}]}=(2n+1+\\ell)\\Bigg[\\frac{e^2B^2}{\\hbar^2}+\\frac{4\\lambda m^2c^2}{\\hbar^2}+\\frac{32\\lambda E^2(\\eta-1)}{\\hbar^2c^2(2\\eta-1)^3}\\nonumber\\\\\n&+\\frac{2\\lambda eB(2\\ell+s)}{\\hbar}+\\frac{4\\lambda eBs}{\\hbar(2\\eta-1)(2+[2\\eta-1]\\frac{mc^2}{E})}\\Bigg]^{\\frac{1}{2}}.\n\\end{align}\nWe may solve this equation for $E$ by keeping again only the first-order terms in the parameter $\\lambda$, to arrive at the following quantization condition on the energy $E$ of \\mbox{the particle}:\n\\begin{align}\\label{FinalQuantization}\nE_{n,\\ell,s}&=\\hbar c\\left(\\eta-\\frac{1}{2}\\right)\\Bigg\\{\\frac{m^2c^2}{\\hbar^2}-\\frac{eB(\\ell+s)}{\\hbar}-\\frac{\\lambda eBs}{4\\hbar}+\\frac{\\lambda(2\\ell^2+s\\ell-2)}{2}\\nonumber\\\\\n&\\quad+\\frac{\\lambda(s\\ell+1)}{(2\\eta-1)}\\frac{\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}{1+\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}+(2n+1+\\ell)\\Bigg[\\frac{e^2B^2}{\\hbar^2}+\\frac{4\\lambda m^2c^2}{\\hbar^2}\\nonumber\\\\\n&\\quad+\\frac{2\\lambda eB(2\\ell+s)}{\\hbar}+(\\eta-1)\\frac{8\\lambda(m^2c^2+[2n+1-s]\\hbar eB)}{\\hbar^2(2\\eta-1)}\\nonumber\\\\\n&\\quad+\\frac{2\\lambda eBs}{\\hbar(2\\eta-1)}\\frac{\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}{1+\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}\\Bigg]^{\\frac{1}{2}}\\Bigg\\}^{\\frac{1}{2}}.\n\\end{align}\n\nWe readily note that for $M=0$ (i.e., by setting $\\eta=\\frac{3}{2}$ and $\\lambda=0$), the result (\\ref{FinalQuantization}) reduces to the usual quantized energy levels of a charged fermion inside a uniform magnetic field in Minkowski spacetime: $E_{n,s}=[m^2c^4+\\hbar c^2(2n+1-s\\frac{e}{|e|})|e|B]^{\\frac{1}{2}}$. In addition, we also note, by setting $B=0$, that even in the absence of the magnetic field the gravitational field induces quantized energy levels. Furthermore, we neatly recognize a general-relativistic correction to the pure relativistic Landau levels arising from the magnetic field. Such a correction arises from the factor $(\\eta-\\frac{1}{2})$ multiplying the right-hand side. \n\nWe are now going to discuss the ways to overcome the limitations we encountered in this section and what modifications would be brought to the results we obtained so far.\n\n\\section{Refinements and Prospective Extensions}\\label{sec:IV}\nTwo limitations, we could not avoid when deriving our results in the previous section, were (i) the fact that we considered only the possible motion of the particle along the equator inside the massive sphere and (ii) the fact that we ignored the geometric effect that might be caused to spacetime by the magnetic field. The two other limitations we imposed on our analysis were (iii) the uniform mass density we assumed for the interior of the spherical mass and (iv) the first-order approximation in $\\lambda$ we limited ourselves to when writing Eq.\\,(\\ref{1stOrderFinalfEq}) to be able to solve the latter exactly. Our aim in this section is to address these limitations and argue that they do not actually alter much the actual physical conclusions we would have reached had we not adopted those various approximations.\n\nThe first limitation consists of the restricted number of degrees of freedom we imposed on the charged fermion. By restricting the motion of the particle to the equatorial plane inside the massive sphere, we implicitly assumed the particle's motion to be confined to such a plane due to the collective interaction of the particles among themselves. However, it is actually possible, for the same reason, to also consider the possibility of motion along a plane other than the equatorial plane. In fact, the vertical pull of gravity is then balanced by these other interactions so that only the horizontal variation of the gravitational potential matters. For such a case, however, we need to switch entirely to cylindrical coordinates. The interior Schwarzschild solution needs to be expressed in such coordinates and we need to set $r^2=\\rho^2+z_0^2$ for some vertical distance $z_0$ from the equator in the resulting differential equation that would be similar in form to Eq.\\,(\\ref{2-SpinorEq}). As the differential operator $\\partial_\\theta$ would be replaced by $z_0\\partial_\\rho$ and the operator $\\partial_\\theta^2$ would be replaced by $z_0^2\\partial_\\rho^2-(\\rho+z_0)\\partial_\\rho$, the equation would still be a second-order differential equation in the single variable $\\rho$. A first-order approximation in the parameter $\\lambda$ might then be adopted. Although such an equation would certainly not be as easy to solve as Eq.\\,(\\ref{1stOrderFinalfEq}), we expect the energy of the particle to be only shifted by a constant while obeying a quantization condition similar to our results (\\ref{QuantizedEpsilon}) and (\\ref{FinalQuantization}). A further refinement consists then of considering the case $\\partial_z\\Psi\\neq0$ and a non-constant $z$. Unfortunately, the fact that two variables would then be involved, no simple procedure could be expected. A separate future work should specifically be devoted to solving that case using cylindrical coordinates. Indeed, the problem of solving our very general Eq.\\,(\\ref{2-SpinorEq}) without restricting the latter to the equatorial plane might be very important for other astrophysical applications besides neutron stars. We believe that the problem might very well be tackled by adapting to our case the existing numerical methods that have already proved fruitful elsewhere for solving such equations (see, e.g., the recent work \\cite{Numerical} and the references therein).\n\nThe second interesting refinement that should be discussed here is the possibility of adding to the metric components, given by Eq.\\,(\\ref{InteriorSchwar}), the spacetime curvature caused by the magnetic field. The geometric effect of combining the external gravitational field sourced by a spherical mass with a uniform magnetic field $B$ gives rise to the so-called Schwarzschild\u2013Melvin spacetime \\cite{Ernst}. Therefore, an analogue modification to the interior Schwarzschild metric components given by Eq.\\,(\\ref{InteriorSchwar}) is also expected when including the spacetime curvature caused by $B$. However, since the corrections brought to the familiar exterior Schwarzschild solution by the magnetic field are of the order of $\\epsilon_0GB^2\/c^2$, where $\\epsilon_0$ is the vacuum permittivity constant \\cite{Galtsov}, we expect the correction that would be brought to the parameter $\\lambda$ in our result (\\ref{FinalQuantization}) to be of the order of $\\sim$$\\epsilon_0GB^2\/c^2$ as well. For a neutron star of mass $\\sim$$1.4\\,M_\\odot$ and of radius $10$\\,km, the Schwarzschild radius is about $\\sim$$4.1\\,$km for which we find that $\\lambda\\sim0.004\\,$km$^{-2}$. On the other hand, for such a neutron star with an interior magnetic field as strong as $\\sim$$10^{17}\\,$G, we have $\\epsilon_0GB^2\/c^2\\sim7\\times10^{-9}\\,$km$^{-2}$. The expected correction is thus quite insignificant.\n\nThe third limitation arises from having discarded in Eq.\\,(\\ref{1stOrderFinalfEq}) terms of the second order and higher in the parameter $\\lambda$. We can justify now such an approximation by the fact that, as we just saw for a typical neutron star, $\\lambda\\sim4\\times10^{-3}\\,$km$^{-2}$. This entails that if we had kept those higher order terms the expected correction to be brought to the quantized energy levels (\\ref{FinalQuantization}) can be obtained using time-independent perturbation theory in the manner worked out in Refs. \\,\\cite{GravityLandauI,GravityLandauII} for the case of Landau levels in the exterior Schwazschild solution. \n\nAnother refinement that should be taken into account is the fact that matter density inside neutron stars is not really uniform as we have assumed it in our calculations in the previous section. The interior matter density increases with increasing depth. However, the core of a neutron star makes up the largest part of the star and may be subdivided into outer and inner parts, each characterized by slowly varying mass densities with depth \\cite{NSDensity}. For this reason, although considering a uniform mass density is only a simplifying model for the interior of a neutron star, we expect that the conclusions that would be derived by assuming a slowly varying $\\rho$-dependent mass density would not differ much from those obtained here. Unfortunately, an analysis based on the interior Schwarzschild solution with a $\\rho$-dependent mass density is beyond the scope of the present paper and should be considered in future works as well. It must also be noted here that numerical methods would certainly be beneficial for such a task.\n\nThe last two extensions of this work we would like to mention here is the possibility of studying the effect of the interior Schwarzschild solution on (i) charged fermions when the magnetic field is nonuniform and (ii) on the neutron star matter when the latter is taken to be a superfluid \\cite{NSSuperfluid}. For the former case, one needs to use again numerical methods (as in Ref.\\,\\cite{NonUniformB}) to solve the resulting equation, whereas for the latter case, one needs to consider, instead of the Dirac equation, the Gross\u2013Pitaevskii equation coupled to the magnetic field inside the interior Schwarzschild solution.     \n\n\\section{Applications}\\label{sec:V}\nDespite the limitations we discussed above, it is still interesting to apply our results to some systems that rely on the Landau energy levels of charged fermions. We will first examine the charged fermions inside a neutron star, and then we examine charged fermions inside a laboratory-made spherical mass.\n\n\\subsection{Inside a Neutron Star}\nWe propose to use here our results (\\ref{QuantizedEpsilon}) and (\\ref{FinalQuantization}) to examine the effect of gravity on the magnetization $\\mathscr{M}$ inside a neutron star.  The magnetization $\\mathscr{M}$ is given by the following general formula \\cite{Broderick}:\n\\begin{equation}\\label{GeneralM}\n    \\mathscr{M}=\\sum_{i=e,\\mu,p}\\left(\\frac{\\partial \\epsilon_i}{\\partial B}-\\mu^i_f\\frac{\\partial n_i}{\\partial B}\\right),\n\\end{equation}\nwhere the summation is over the electrons, muons and protons, respectively; $\\epsilon_i$ are the energy densities of those charged fermions and $n_i$ are their number densities. The magnetic field $B$ in this formula is the original seed field (magnetizing field) that is thus gradually modified by the magnetization into $H=B+4\\pi\\mathscr{M}$, in CGS units \\cite{LLPTextbook}. A strong magnetization has been suggested by some authors to play a role even in giving rise to the original (primal) magnetic field itself \\cite{Dong2013}. However, despite considerable research on the topic, there is in the literature no consensus yet on the origin of such strong seed magnetic fields at the surface nor in the core of such compact stars. For a quick guide through the various proposals put forward in the literature, see, e.g., the reviews \\cite{ReviewEoS2015,Universe2021}. \n\nConsidering the magnetizing field $B$ to be constant in time and uniform over the radius of the star, as we do in this paper, is also only an approximation. However, given that the relaxation times of the star's magnetic field decay is of the order of $\\sim$$10^3$ \\mbox{years \\cite{Universe2018}}, our present assumption of a constant magnetic field fits amply within the early stages of the magnetic evolution of the star. Similarly, assuming a slow variation of the magnetic field with distance inside highly magnetized stars as presented, for example, in Ref.\\,\\cite{BDistribution} is still consistent with our present purpose of providing a preliminary study on the effect of gravity on the magnetic properties of such compact stellar objects.\n\nFirst, recall that when ignoring the contribution of the gravitational field, the quantities $\\epsilon_i$ and $n_i$ are given by \\cite{Broderick}\n\\begin{align}\\label{EnergyAndNumberDensity}\n    \\epsilon_i&=\\frac{|e|B}{4\\pi^2}\\sum_s\\sum_{n=0}^{n_{max}}\\left(\\mu^i_fk^i_{f,n,s}+\\tilde{m}^{i2}_{n,s}\\ln\\left|\\frac{\\mu^i_f+k^i_{f,n,s}}{\\tilde{m}^i_{n,s}}\\right|\\right),\\nonumber\\\\\n    n_i&=\\frac{|e|B}{2\\pi^2}\\sum_s\\sum_{n=0}^{n_{max}}k^i_{f,n,s},\n\\end{align}\nwhere\n\\begin{align}\\label{mTilde}\n    \\tilde{m}^{i2}_{n,s}&=m^{i2}+\\left(2n+1-\\frac{e}{|e|}s\\right)|e|B,\\nonumber\\\\\n    k_{f,n,s}^{i2}&=\\mu_f^{i2}-\\tilde{m}_{n,s}^{i2}.\n\\end{align}\nThe Fermi energies $\\mu^{i}_f$ are fixed by the chemical potentials and the maximum integer $n_{max}$ in the summations represents the integer preceding the value of $n$ for which $k_{f,n,s}^{i2}$ becomes negative. As the equations are more involved, we will work in this subsection in the natural units $\\hbar=c=1$. Formula (\\ref{GeneralM}) then gives \\cite{Broderick} \\begin{equation}\\label{FlatMagnetization}\n    \\mathscr{M}=\\sum_{i=e,\\mu,p}\\left[\\frac{\\epsilon_i-\\mu^i_fn_i}{B}+\\frac{B}{2\\pi^2}\\sum_{s}\\sum_{n=0}^{n_{max}}\\left(n+\\frac{1}{2}-\\frac{s}{2}\\right)\\ln\\left|\\frac{\\mu^i_f+k^i_{f,n,s}}{\\tilde{m}^i_{n,s}}\\right|\\right].\n\\end{equation}\n\nWe will adapt these general formulas to our case. Since the formulas to be developed here are more relevant for very strong magnetic fields, we also extract from Eq.\\,(\\ref{FinalQuantization}) the following approximation for the quantized energy levels:\n\\begin{align}\\label{StrongBApproximation}\n    E_{n,\\ell,s}&=\\left(\\eta-\\frac{1}{2}\\right)\\Bigg\\{m^2\\!+\\!\\left(2n+2\\ell+1-\\frac{e}{|e|}s\\right)|e|B\\!-\\!\\frac{\\lambda eBs}{4\\hbar}\\!+\\!\\frac{\\lambda(2\\ell^2+s\\ell-2)}{2}\\!+\\!\\frac{\\lambda(s\\ell+1)}{2\\eta-1}\\nonumber\\\\\n&\\quad+(2n+1+\\ell)\\Bigg[\\lambda (2\\ell+s)+(\\eta-1)\\frac{4\\lambda(2n+1-s)}{2\\eta-1}+\\frac{\\lambda s}{2\\eta-1}\\Bigg]\\Bigg\\}^{\\frac{1}{2}}.\n\\end{align}\nNote that in the absence of gravity, i.e., when $\\lambda=0$, this formula reduces to the first identity in Eq.\\,(\\ref{mTilde}). Let us then start with the consequences of this formula which is valid either (i) for weak gravitational fields and\/or (ii) for fermions very close to the center of the star. Since the terms proportional to $\\lambda$ in this expression de not contain $B$, it is straightforward to compute the $B$-derivatives in Eq.\\,(\\ref{GeneralM}). The result for the magnetization remains thus exactly the same as in Eq.\\,(\\ref{FlatMagnetization}), but with $\\tilde{m}_{n,s}^{i2}$ replaced by $E_{n,\\ell,s}$ as given by Eq.\\,(\\ref{StrongBApproximation}):\n\\begin{equation}\\label{CurvedMagnetization}\n    \\mathscr{M}=\\sum_{i=e,\\mu,p}\\left[\\frac{\\epsilon_i-\\mu^i_fn_i}{B}+\\frac{B(2\\eta-1)}{4\\pi^2}\\sum_{s}\\sum_{n=0}^{n_{max}}\\left(n+\\frac{1}{2}-\\frac{s}{2}\\right)\\ln\\left|\\frac{\\mu^i_f+k^i_{f,n,s}}{E^i_{n,\\ell,s}}\\right|\\right].\n\\end{equation}\nWe neatly see the general-relativistic correction factored out in the second term of the sum. More important, however, is the fact that the {\\it general expression} of the magnetization remains unaltered in this case.\n\nOn the other hand, for the case of a {\\it non-negligible} gravitational field the quartic term in Eq.\\,(\\ref{1stOrderFinalfEq}) should be taken into account, leading to the result (\\ref{QEpsilonN0}) for $E_{n,\\ell,s}$ at the first-order term of the JWKB approximation. As the Landau levels are completely destroyed, we see that even at this order in the approximation the magnetization is completely different from the expression (\\ref{CurvedMagnetization}). Actually, even Formulas (\\ref{EnergyAndNumberDensity}) and (\\ref{mTilde}) do not hold anymore, for such formulas were derived by assuming the linear dependence (\\ref{mTilde}) of $\\tilde{m}_{n,s}^{i2}$ on the integer $n$. As expression (\\ref{QEpsilonN0}) of $E_{n,\\ell,s}$ in terms of $n$ displays no such linearity, and since that expression (\\ref{QEpsilonN0}) is itself only an approximation for the highly nonlinear full \\mbox{expression (\\ref{QuantizedEpsilon}),} we conclude that the Landau-levels-based formalism for computing the magnetization $\\mathscr{M}$ of neutron stars does not generally apply to the charged plasma in the core of highly dense neutron stars. The condition for the applicability of the formalism is to have the quartic anharmonic oscillator potential term in Eq.\\,(\\ref{Anharmonic}) negligible, i.e., to have $\\lambda\\ll |e|B\/\\hbar$ or to apply the formalism only to fermions close to the vicinity of the star's center. In other words, we conclude that only for a magnetic neutron star of mass $M$, of radius $R$ and with a magnetic field $B$ such that $|e|BR^3c^2\\gg GM\\hbar$, that the usual formalism generally applies to the charged matter inside the core of the star.\n\nMagnetization in neutron stars affects both the equation of state of the stars' matter and the stars' structure and properties, such as their masses and transport properties. Indeed, both the pressure anisotropy and the softening of the equation of state of the stars matter is believed to be related to magnetization and the distribution of the charged fermions as they occupy the Landau levels. The equation of state determines, in turn, the mass-radius relation of neutron stars (see, e.g., the recent work \\cite{EOS2020} and the \\mbox{references therein}).  \n\nIt is believed that the net magnetization in neutron stars remains negligible until around $10^{16}$\\,G, beyond which it starts increasing and becomes oscillatory under the de Haas-van Alphen effect. Since we just saw that the Landau levels become dramatically altered only for non-negligible gravitational fields compared to the magnetic interaction, we conclude that such oscillations will not be much affected in typical stars. For weaker magnetic fields, however, the general-relativistic correction we found in Eq.\\,(\\ref{CurvedMagnetization}) for the magnetization might leave an observable signature on the equation of state of the star's matter and, hence, on the mass-radius relation of the star as well. More work dedicated specifically to such an interesting investigation is required though.\n\\subsection{At a Laboratory Level}\nAnother application of our results is to consider physical systems at the laboratory level. In this case, an interesting system would be the free electrons moving inside a massive metallic sphere put inside a uniform magnetic field. For this case, the gravitational contribution is much weaker than the magnetic interaction, so Eq.\\,(\\ref{FinalQuantization}) is what we need to apply here. Expanding that expression up to the first order in $\\lambda$, we obtain\n\\begin{align}\\label{FinalQuantizationWeakG}\nE_{n,\\ell,s}&\\approx\\left(\\eta-\\frac{1}{2}\\right)\\Bigg\\{m^2c^4+\\hbar c^2\\left(2n\\!+\\!2\\ell\\!+\\!1\\!-\\!\\frac{e}{|e|}s\\right)|e|B-\\frac{\\hbar c^2\\lambda eBs}{4}+\\frac{\\hbar^2 c^2\\lambda(2\\ell^2\\!+\\!s\\ell\\!-\\!2)}{2}\\nonumber\\\\\n&\\quad+\\frac{\\lambda\\hbar^2c^2(s\\ell+1)}{(2\\eta-1)}\\frac{\\sqrt{1+(2n\\!+\\!1\\!-\\!s)\\frac{\\hbar eB}{m^2c^2}}}{1+\\sqrt{1+(2n\\!+\\!1\\!-\\!s)\\frac{\\hbar eB}{m^2c^2}}}+(2n+1+\\ell)\\hbar^2c^2\\lambda\\Bigg[\\frac{2 m^2c^2}{\\hbar eB}+2\\ell+s\\nonumber\\\\\n&\\quad+(\\eta-1)\\frac{4(m^2c^2+[2n+1-s]\\hbar eB)}{\\hbar eB(2\\eta-1)}+\\frac{ s}{2\\eta-1}\\frac{\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}{1+\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}\\Bigg]\\Bigg\\}^{\\frac{1}{2}}.\n\\end{align}\nThese are the energy levels of the electrons inside the massive sphere. The $\\lambda$-terms are the corrections brought to the familiar levels observed in the absence of gravity. This expression is valid for both relativistic and non-relativistic fermions, and it reduces to the familiar relativistic Landau levels in the absence of gravity. This result offers novel ways for exploiting the Landau levels of fermions in the presence of gravity at the \\mbox{laboratory level.}\n\n\\section{Summary}\\label{sec:VI}\nWe have obtained the Dirac equation for a charged fermion in a general static and spherically symmetric curved spacetime in the presence of a static and uniform magnetic field. We applied the general equation we obtained to the case of fermions in the {\\it interior} Schwarzschild solution describing the gravitational field inside a massive sphere of uniform density, and we derived the quantized energy levels of the charged particles. We found that our result reduces to the familiar flat-spacetime relativistic Landau levels of charged fermions inside a uniform magnetic field only when the gravitational interaction is weaker than the magnetic interaction and\/or when the charged particles move very closely to the center of the spherical mass. \n\nWe then discussed, based on these results, the consequences on the physics of neutron stars. We found that the magnetization of the core of the latter would dramatically be altered for an extremely high gravitational field compared to the magnetic interaction and\/or when not focusing solely on the fermions in the vicinity of the center of the star. We arrived at the conclusion that in the general case, the usual formalism for extracting the magnetization of neutron stars applies only when the latter satisfy the condition $|e|BR^3c^2\\gg GM\\hbar$. Although we arrived at such a conclusion by relying on a first-order approximation in the gravitational parameter $\\lambda$, the generality of our results is not affected. In fact, such a first-order approximation remains valid for any relative strengths of the gravitational and magnetic interactions experienced by the charged particles inside typical neutron stars. Any extra correction to those levels would be obtained simply using time-independent perturbation theory.\n\nIn Section \\ref{sec:IV}, we outlined a few refinements that might be brought to our model and the prospective outlook on future improvements that will allow us to go beyond the limitations we imposed on our present study. However, even with the simple model dealt with here, we have already glimpsed with new insights into the contribution of gravity in shaping the magnetization of highly compact neutron stars and into novel ways of exploiting the dynamics of charged fermions in the presence of gravity at the laboratory level. \n\n\n\n\n\\section*{Acknowledgments}\nThe authors are grateful to Patrick Labelle for the helpful discussions and comments, and to the anonymous referees for their constructive comments that helped enrich our presentation. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant No. RGPIN-2017-05388 and by the Fonds de Recherche du Qu\u00e9bec - Nature et Technologies (FRQNT). PS acknowledges support from Bishop's University via the Graduate Entrance Scholarship award.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}Minimal surfaces of general type with $p_g(S)\n= 0$ have invariants\n$p_g(S)  =  q (S)= 0, 1 \\leq K_S^2  \\leq 9$, and, for each value of $\ny \\in \\{1,2, \\dots, 9\\}$, such surfaces\nwith $ K_S^2 = y$  yield  a finite number\nof irreducible components of the Gieseker moduli space of surfaces of\ngeneral type $\\mathfrak M^{can}_{1,y}$.\n\nAfter the first surfaces of general type with $p_g = q = 0$ were\nconstructed in the 1930' s by   Luigi Campedelli and by Lucien\nGodeaux (cf. \\cite{Cam},\n\\cite{god}) there was  in the 1970's a big revival of interest in the\nconstruction of these surfaces,\nas documented by a long and influential survey paper  written by\nDolgachev (\\cite{Dolgachev}). \n\nThe Bloch conjecture and\ndifferential topological questions raised by Donaldson Theory were a\nfurther reason for the remarkable and ongoing  interest\nabout surfaces of general type with\n$p_g = 0$, and  we refer to\n\\cite{survey} for an update  about   recent  important progress on\nthe topic, and about  the state of the art.\n\n\nLooking at the tables 1--3 of \\cite{survey} one finds  it striking\nthat for the value $K_S^2= 7$ there is only one  known family of\nsuch surfaces of general type. This family was constructed by Inoue\n(cf. \\cite{inoue}).\nFurther interest concerning this family comes from the problem raised\nin \\cite{survey}, and concerning the finiteness or not\nof the fundamental group of  surfaces with $p_g = 0$. Indeed this paper was\nmotivated by the observation that Inoue's surfaces have a\n``big'' fundamental group. In fact, the fundamental group of an Inoue\nsurface with $p_g = 0$ and $K_S^2 =7$ sits\nin an extension ($\\pi_g$ denotes the fundamental group of a compact\ncurve of genus $g$):\n$$ 1 \\rightarrow \\pi_{5} \\times \\mathbb{Z}^4 \\rightarrow \\pi_1(S)\n\\rightarrow (\\mathbb{Z}\/2\\mathbb{Z})^5 \\rightarrow 1.\n$$\n\nThis extension is given geometrically, i.e., stems from our observation that\nan Inoue surface $S$ admits an unramified\n$(\\mathbb{Z}\/2\\mathbb{Z})^5$ - Galois covering $\\hat{S}$ which is an\nample divisor in $E_1 \\times E_2 \\times D$, where\n$E_1, E_2$ are elliptic curves and $D$ is a compact curve of genus\n$5$; this made us believe\n   that the topological type of an Inoue surface determines an irreducible\nconnected component of the moduli space (a phenomenon similar to the\none which was already observed   in \\cite{keumnaie},\n\\cite{burniat1} \\cite{coughlanchan}).\n\nThe following is  one of the main  results of this paper:\n\n\\begin{theo}\\label{main}\\\n\\begin{enumerate}\n\\item Let $S'$ be a smooth complex projective surface which is\nhomotopically equivalent to an Inoue surface (with $K^2 = 7$ and\n$p_g=0$). Then $S'$ is an Inoue  surface.\n\\item The connected component of the Gieseker moduli space\n$\\mathfrak M^{can}_{1, 7}$ corresponding to Inoue surfaces is\nirreducible, generically smooth, normal and unirational  of dimension  4.\n\nMoreover, each Inoue surface $S$ has ample canonical divisor\\footnote{This is proven by Inoue, \nsee page 318 of \\cite{inoue}.} and the base $\\Def (S)$  of the Kuranishi family of $S$  is smooth.\n\\item\nFinally, the first homology group of an Inoue surface equals $$\\ensuremath{\\mathbb{Z}}\/4 \\ensuremath{\\mathbb{Z}} \\oplus (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4.$$\n\n\\end{enumerate}\n\\end{theo}\n\nSince this theorem is similar in flavour to other results that we \nmentioned above,\nthe main purpose of this paper is not only to give a more general proof, but also to  set up the\nstage for  the investigation and search for a new class of varieties, \nwhich we propose to call\nInoue-type varieties.\n\n\\begin{defin}\nWe define a complex projective manifold $X$ to be an {\\bf Inoue-type manifold} if\n\\begin{enumerate}\n\\item\n$ dim (X) \\geq 2$;\n\\item\nthere is a finite group $G$ and a Galois unramified covering $\n\\hat{X} \\ensuremath{\\rightarrow} X$ with group $G$,\n(hence $ X = \\hat{X} \/ G$) so that\n\\item\n$ \\hat{X}$ is an ample divisor inside a $K(\\Ga, 1)$-projective manifold\n$Z$ (hence by Lefschetz $\\pi_1 ( \\hat{X}) \\cong \\pi_1 (Z)\n\\cong \\Ga$) and moreover\n\\item\nthe action of $G$ on $ \\hat{X}$   yields a faithful action on $\\pi_1\n( \\hat{X}) \\cong \\Ga$:\nin other words the exact sequence\n$$ 1 \\ensuremath{\\rightarrow} \\Ga  \\cong \\pi_1 ( \\hat{X}) \\ensuremath{\\rightarrow} \\pi_1 ( X) \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1$$\ngives an injection $ G \\ensuremath{\\rightarrow} Out (\\Ga)$, defined by conjugation;\n\\item\nthe action of $G$ on $ \\hat{X}$   is induced by an action on $Z$.\n\\end{enumerate}\nSimilarly one defines the notion of an  {\\bf Inoue-type variety}, by\nrequiring the same properties\nfor a variety $X$ with canonical singularities.\n\\end{defin}\n\n\nWe should warn the reader that our approach was inspired by, but is different from the\noriginal construction of Inoue, who considers\nhypersurfaces and complete intersections (of not necessarily ample\ndivisors) in a product of elliptic curves.\nIn fact, the change of point of view in the particular case  of  an Inoue surface with \n$K^2 = 7, p_g = 0$ produces a different realization: instead of Inoue's   original\nrealization as\n a complete intersection of two hypersurfaces of multidegrees $(2,2,2,0)$ and\n$(0,0,2,2)$ in a product of 4 elliptic curves,\n  we view the same surface as a hypersurface of multidegrees\n$(2,2,4)$ in the product\n$E_1 \\times E_2 \\times D$ of two elliptic curves with a curve $D$ of genus $5$.\n\nOne can see that our definition, although imposing a strong restriction on $X$, \nis not yet satisfactory in order to obtain some\nweak rigidity result (of the type of theorems 4.13 and 4.14 of \n\\cite{isogenous}, amended in \\cite{annals}, theorem 1.3).\nSome hypotheses must be made on the fundamental group $\\Ga$ of $Z$,\nfor instance the most interesting case\nis the one where $Z$ is a product of Abelian varieties, curves, and other locally symmetric varieties with\nample canonical bundle.\n\n\\begin{defin}\\label{SIT}\n   We shall say  that an  Inoue-type manifold $X$  is \n\n\\begin{enumerate}\n\\item\n  a {\\bf SIT : = special Inoue type manifold}\nif  moreover\n$$ Z = (A_1 \\times \\dots \\times A_r) \\times  (C_1 \\times \\dots \\times\nC_h) \\times (M_1 \\times \\dots \\times\nM_s)$$ where each\n$A_i$ is an Abelian variety,  each $C_j$ is a curve of genus $ g_j \\geq 2$,\nand $M_i$ is  a\ncompact quotient of an irreducible bounded symmetric domain of \ndimension at least 2 by a\ntorsion free subgroup.\n\\item\n  a {\\bf CIT : = \nclassical Inoue type manifold}\nif  moreover\n\n$ Z = (A_1 \\times \\dots \\times A_r) \\times  (C_1 \\times \\dots \\times\nC_h) $ where each\n$A_i$ is an Abelian variety,  each $C_j$ is a curve of genus $ g_j \\geq 2$.\n\\item\na special Inoue type manifold  is said to be\na \n\n{\\bf diagonal SIT manifold : = diagonal\nspecial   Inoue type manifold }\nif  moreover:\n\\begin{itemize}\n\\item the action of $G$ on $ \\hat{X}$   is induced by a  diagonal \naction on $Z$, i.e.,\n\n$$\n(I) \\ \\  G \\subset    \\prod_{i=1}^r \\Aut(A_i)\n\\times\n\\prod_{j=1}^h\\Aut(C_j) \\times \\prod_{l=1}^s\\Aut(M_l)\n$$\n\n\\item\nand furthermore:\n\n(II) \nthe  faithful action on $\\pi_1 ( \\hat{X}) \\cong \\Ga$,\ninduced by conjugation by lifts of elements of $G$ in  the exact sequence\n\\begin{equation}\\label{prodFG}\n 1 \\ensuremath{\\rightarrow} \\Ga= \\Pi_{i=1}^r (\\Lambda_i) \\times   \\Pi_{j=1}^h (\\pi_{g_j}) \n\\times \\Pi_{l=1}^s\n(\\pi_1 (M_l))\n\\ensuremath{\\rightarrow} \\pi_1 ( X) \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1\n\\end{equation}\n(observe that each factor $\\Lambda_i$, resp. $\\pi_{g_j}, \\pi_1 (M_l)$ is \nnormal), has the Schur property\n$$(SP)  \\Hom (V_i, V_j)^{G} = 0,\\forall i \\neq j,   $$\nwhere $V_j : = \\Lambda_j\\otimes \\ensuremath{\\mathbb{Q}}$ (it  suffices then to verify  that \nfor each $\\Lambda_i$ there is a subgroup $H_i$ of $G$ for which $ \\Hom (V_i, V_j)^{H_i} = 0, \\forall j \\neq i  $).\n\\end{itemize}\n\\item\nsimilarly we define a {\\bf diagonal CIT manifold : = diagonal\nclassical  Inoue type manifold }\n\n\\end{enumerate}\nWe can define analogous  notions for    Inoue  type varieties\n  $X$ with canonical singularities.\n\\end{defin}\n\n\nProperty (SP) plays an important role in order to show that an\nAbelian variety  with such a $G$-action on its fundamental group\nmust split as a product.\n\nThere is however  a  big difference between the curve and locally symmetric\nfactors  on one side and the\nAbelian variety factors on the other.  Namely: for curves we have weak rigidity, i.e., \nthe action of $G$ on $\\pi_{g_j}$ determines a connected family of\ncurves; for compact free quotients of bounded symmetric domains of \ndimension $\\geq 2$ we have strong rigidity, i.e., \nthe action on the fundamental group determines uniquely the \nholomorphic action; \nfor Abelian varieties it is not necessarily so.\n\nHence, in order to hope for weak rigidity results, one has to \nintroduce  a further invariant, called Hodge type (see section 1).\n\nWe can now state our main general result:\n\n\n\\begin{theo}\\label{special diagonal}\\\n  Let $X$ be a diagonal SIT (special  Inoue type) manifold, and let $X'$ be a\nprojective manifold\nwith the same fundamental group as $X$, which moreover either\n\n(1)\nis homotopically equivalent to $X$;\n\n  or satisfies the following weaker property:\n\n(2) let $\\hat{X'}$ be the corresponding unramified covering of $X'$ and\nidentify both\nfundamental groups $\\pi_1 (\\hat{X'}) = \\pi_1 (\\hat{X})= \\Ga $ . Then \nwe require that\n\n {\\bf (SAME HOMOLOGY)} there is an isomorphism $\\Psi : H_*(\\hat{X'}, \\ensuremath{\\mathbb{Z}}) \\cong\nH_*(\\hat{X}, \\ensuremath{\\mathbb{Z}})  $\nof homology groups which is compatible with the homomorphisms\n$$ u \\colon H_*(\\hat{X}, \\ensuremath{\\mathbb{Z}})  \\ensuremath{\\rightarrow}   H_*(\\Ga , \\ensuremath{\\mathbb{Z}})   ,  u'   \\colon\n H_*(\\hat{X'}, \\ensuremath{\\mathbb{Z}})  \\ensuremath{\\rightarrow}   H_*(\\Ga , \\ensuremath{\\mathbb{Z}})  ,   $$\ni.e., $u \\circ \\Psi = u'$.\n\n\nUnder these assumptions, we have, setting $ W : = \\hat{X'}$, that\n\n\\begin{itemize}\n\\item\n$X'=  W \/ G$ where $W$ admits a generically finite\nmorphism \n$f : W \\ensuremath{\\rightarrow} Z'$, and where\n\\item\n$Z'$ is also a $K(\\Ga, 1)$ projective manifold,\nof the form $ Z' = (A'_1 \\times \\dots \\times A'_r) \\times\n(C'_1 \\times \\dots \\times  C'_h) \\times (M'_1 \\times \\dots \\times\nM'_s)$.\n\nMoreover here $M'_i$ is either $M_i$ or its complex conjugate,\nand the product decomposition corresponds to the product decomposition\n(\\ref{prodFG}) of the fundamental group of $Z$.\n\\item\nThe  image cohomology class  $f_*([W])$ corresponds, up to sign,\nto the cohomology class of $\\hat{X}$.\n\\item\nThe morphism $f$ is\nfinite if $n = dim X$ is odd, and moreover it is\ngenerically injective if \n\n(**) the cohomology class of $\\hat{X}$ is indivisble,\nor if every strictly submultiple  cohomology class cannot be represented by an effective\n$G$-invariant divisor  on any pair  $(Z', G)$ homotopically equivalent to $(Z,G)$.\n\\item\n $f$ is an embedding if moreover $K_{X'}$ is ample and\n\n(***)  $\\ \\ \\ K_{X'}^n = K_{X}^n$.\\footnote{ This last property for\nalgebraic surfaces follows automatically from homotopy invariance.}\n\\end{itemize}\nIn particular, if $K_{X'}$ is ample and (**) and (***) hold, also $X'$ is a diagonal SIT (special  Inoue type) manifold.\n\nA similar  conclusion holds under the alternative assumption that  the isomorphism of homology groups sends the canonical class of $W$\nto the one of $\\hat{X}$: then  $X'$ is a diagonal SIT (special  Inoue type) variety.\n\n\\end{theo}\n\nHypothesis  (1) in the previous theorem allows to derive the\nconclusion that also\n$W : = \\hat{X'} $ admits a holomorphic map $f'$ to a complex manifold $Z'$\nwith the same structure as $Z$, while\n hypotheses  (2) and following ensure  that the morphism is birational to its image,\nand the class of the image\ndivisor $f' (\\hat{X'} )$ corresponds to $\\pm$ the one of $\\hat{X} $  under\nthe identification\n$$ H_*(Z' , \\ensuremath{\\mathbb{Z}}) \\cong  H_*(\\Ga , \\ensuremath{\\mathbb{Z}}) \\cong   H_*(Z , \\ensuremath{\\mathbb{Z}}).$$\n\nSince $K_{X'}$ is ample, one uses (***) to conclude that $f'$ is an isomorphism with its image.\n\nThe next question is weak rigidity, which amounts to the existence of a\nconnected complex manifold\nparametrizing all such maps (or the  complex conjugate).\nHere  several ingredients come into play, namely, firstly the Hodge type,\nsecondly a fine analysis of\nthe structure of the action of $G$ on $Z$, in particular concerning the existence of hypersurfaces on which $G$ acts freely. Finally, one would have to see\nwhether the family of the invariant effective divisors\nthus obtained is parametrized by a connected family: this also\nrequires further work which we do not undertake here except that for the case of Inoue surfaces.\n\nIt would  take  long to analyse here the most general situation, yet\n there is an even  more general\nsituation worth to be investigated. This is the case of {\\bf orbifold\nInoue type varieties},\nwhere the action of $G$ is no longer free: this situation is\nespecially appealing from the point of view of the construction of new\ninteresting examples.\n\n\nThe paper is organized as follows:  in the first section we deal with general Inoue type manifolds,\nestablish the first  general properties\nof Inoue type manifolds, and prove our main theorem \\ref{special diagonal}. Further, more complete, results\ndealing with weak rigidity will be given elsewhere.\n\nSection two is devoted to preliminaries, for instance  on curves of \ngenus 5 admitting symmetries by  $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4$.\nThis is an important background for the construction of Inoue \nsurfaces with $K^2_S = 7$ and $p_g (S)= 0$,\nwhich is explained in detail in section three.\nThe end of section three is then devoted to a new result, namely, the calculation of the\nfirst homology group of an Inoue surface, which is shown to be equal to the  group $(\\ensuremath{\\mathbb{Z}}\/4 \\ensuremath{\\mathbb{Z}})\\oplus (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4$.\n\nSection four proves the main result on Inoue surfaces with $K^2_S = \n7$ and $p_g (S)= 0$.\n\nFinally, section five is devoted to showing that the the moduli space \nof Inoue surfaces is generically smooth: this is achieved\nby looking at another  realization as  bidouble covers of a four nodal cubic.\n\\section{ Inoue type manifolds and varieties}\n\nAssume that $X = \\hat{X} \/ G$ is an Inoue-type manifold, so that \nthere is an isomorphism\n$\\pi_1 (\\hat{X} ) \\cong \\pi_1 (Z )= : \\Ga $, by virtue of Lefschetz' theorem.\n\nIn general, if $W$ is another K\\\"ahler manifold with $\\pi_1 (W  ) \n\\cong \\pi_1 (Z )= : \\Ga $,\nwe would like to be able to assert that there exists a holomorphic \nmap $ f : W \\ensuremath{\\rightarrow} Z'$ where $Z'$ is another $K(\\Ga,1)$ manifold and \nwhere  $f_* : \\pi_1 (W  ) \\ensuremath{\\rightarrow} \\pi_1 (Z' ) \\cong \\Ga$ realizes the above isomorphism.\n\n This is for instance the case if $Z$ is a\ncompact quotient of an irreducible bounded symmetric domain of \ndimension at least 2 by a\ntorsion free subgroup; this follows by combining the results of Eells and Sampson  (\\cite{es})\nproving the existence of a harmonic map in each homotopy class of maps $ f : W \\ensuremath{\\rightarrow} Z$, since $Z$\nhas negative curvature, with the results of Siu (\\cite{siu1} and \n\\cite{siu2}), showing the complex dianalyticity of the resulting harmonic map  (i.e., the map $f$  is holomorphic or antiholomorphic)\nin the case where $f_* : \\pi_1 (W  ) \\ensuremath{\\rightarrow} \\pi_1 (Z' ) \\cong \\Ga$ is an isomorphism, since then the differential of $f$ has rank $\\geq 4$\n(as a linear map of real vector spaces).\nObserve that in this case $Z'$ is  $ Z$, or the complex conjugate $\\bar{Z}$.\n \nAnother case is the case where $Z$ is a compact curve of genus $g \n\\geq 2$. In this case,\nafter several results by Siu, Beauville and others (see \\cite{albgentype}), a\n  simple criterion was shown to be the existence of a surjection $\\pi_1 (W  ) \\ensuremath{\\rightarrow} \n\\pi_1 (Z )$  with finitely generated kernel (see\n\\cite{barlotti} and \\cite{cime2}, Theorem 5.14), an assumption which holds true in our situation.\n\nIt is on the above grounds that we restricted ourselves to special Inoue type manifolds\n(the diagonality assumption is only a simplifying assumption).\n\n\n\nLet us prove the first general result, namely, theorem \\ref{special diagonal}.\n\n{\\it Proof. } {\\em of Theorem \\ref{special diagonal}.}\n\n{ \\bf Step 1}\n\nThe first step consists in showing that  $W : = \\hat{X'}$ admits a holomorphic\nmapping to a manifold $Z'$ of the above type $ Z' = (A'_1 \\times \n\\dots \\times A'_r) \\times\n(C'_1 \\times \\dots \\times  C'_h) \\times (M'_1 \\times \\dots \\times\nM'_s)$, where $M'_i$ is either $M_i$ or its complex conjugate.\n\nFirst of all, by the cited results of Siu and others (\\cite{siu1}, \n\\cite{siu2},\\cite{barlotti},  \\cite{cime2}, Theorem 5.14), $W$ admits a \nholomorphic map to\na product manifold $$Z_2'\\times Z'_3 = (C'_1 \\times \\dots \\times \nC'_h) \\times (M'_1 \\times\n\\dots\n\\times  M'_s).$$\n\nLook now at the  Albanese variety $\\Alb (W)$ of the K\\\"ahler manifold \n$W$, whose fundamental\ngroup is the quotient of the Abelianization of $\\Ga = \\pi_1 (Z)$ by its torsion subgroup. \n\nWrite the fundamental group of\n$\\Alb (W)$ as the first homology group of $A  \\times  Z_2  \\times Z_3$,\ni.e., as \n$$ H_1 ( \\Alb (W)) = \\Lambda \\oplus H_1 ( Z_2, \\ensuremath{\\mathbb{Z}}) \\oplus (H_1 ( Z_3, \\ensuremath{\\mathbb{Z}})\/ \\Torsion),$$ \n($\\Alb (Z_2)$ is the product of Jacobians\n$J : = (\\Jac(C_1) \\times \\dots \\times  \\Jac(C_h))  $).\n\nSince however, by the universal property,  $\\Alb (W)$ has a holomorphic map to \n$$B' : = \\Alb (Z'_2) \\times \\Alb (Z'_3),$$\ninducing a splitting of the lattice  $H_1 (\\Alb (W), \\ensuremath{\\mathbb{Z}}) = \\Lambda \\oplus \nH_1 (B', \\ensuremath{\\mathbb{Z}})$,\nit follows that $\\Alb(W)$ splits as $A' \\times B'$.\n\nNow, we want to show that the Abelian variety $A'$ ($W$ is assumed to \nbe a projective\nmanifold) splits as desired.  This is in turn a consequence of assumption (3) in definition \\ref{SIT}.\nIn fact, the group $G$ acts on the Abelian variety $A'$ as a group of \nbiholomorphisms,\nhence it acts on $\\Lambda \\otimes \\ensuremath{\\mathcal{R}}$ commuting with multiplication by \n$\\sqrt{-1}$.\nHence multiplication by $\\sqrt{-1}$ is an isomorphism of $G$ representations, and then (3)\n implies  that $\\Lambda_i\\otimes \\ensuremath{\\mathcal{R}}$ is stable by  multiplication by \n$\\sqrt{-1}$;\nwhence $\\Lambda_i\\otimes \\ensuremath{\\mathcal{R}}$ generates a subtorus $A'_i$. Finally, $A'$ splits because $\\Lambda$ is\nthe direct sum of the sublattices $\\Lambda_i$. We are through with the \nproof of step 1.\n\n{ \\bf Step 2}\n\nConsider now the holomorphic map $ f \\colon W \\ensuremath{\\rightarrow} Z'$. We shall show \nthat  the image\n$ W' : = f (W)$ is indeed a divisor in $Z'$.\n\nThis follows from the assumption  (SAME HOMOLOGY).\n\n\\begin{lemma}\\label{cohalg}\nAssume that $W$ is a K\\\"ahler manifold, such that\n\\begin{itemize}\n\\item[i)] there is an isomorphism of\nfundamental groups $\\pi_1 (W) = \\pi_1 (\\hat{X})= \\Ga$, where $ \n\\hat{X}$ is a smooth ample\ndivisor in a $K(\\Ga, 1)$ complex projective manifold $Z$;\n\\item[ii)] there exists a holomorphic map  $ f : W \\ensuremath{\\rightarrow} Z'$, where $Z'$ is another \n$K(\\Ga, 1)$ complex  manifold,  such that $f_* : \\pi_1 (W) \\ensuremath{\\rightarrow}  \\pi_1 (Z') = \\Ga$ is an \nisomorphism, and moreover\n\\item[iii)]  {\\bf (SAME HOMOLOGY)} there is an isomorphism $\\Psi : H_*( W, \\ensuremath{\\mathbb{Z}}) \\cong\nH_*(\\hat{X}, \\ensuremath{\\mathbb{Z}})  $\nof homology groups which is compatible with the homomorphisms\n$$ u \\colon H_*(\\hat{X}, \\ensuremath{\\mathbb{Z}})  \\ensuremath{\\rightarrow}   H_*(\\Ga , \\ensuremath{\\mathbb{Z}})   ,  u'   \\colon\n H_*( W, \\ensuremath{\\mathbb{Z}})  \\ensuremath{\\rightarrow}   H_*(\\Ga , \\ensuremath{\\mathbb{Z}})  ,   $$\ni.e., $u \\circ \\Psi = u'$.\n\n\\end{itemize}\nThen $f$ is a generically finite  morphism of $W$ into $Z'$, and \nthe  cohomology class  $f_* ([W])$  in\n$$ H^*(Z', \\ensuremath{\\mathbb{Z}}) = H^*(Z, \\ensuremath{\\mathbb{Z}})\n=H^*(\\Ga, \\ensuremath{\\mathbb{Z}})  $$ corresponds to $\\pm 1$  the one  of $ \\hat{X}$.\n\\end{lemma}\n\n{\\it Proof of the Lemma.}\n We can identify under $\\Psi : H_*(W, \\ensuremath{\\mathbb{Z}}) \\cong\nH_*(\\hat{X}, \\ensuremath{\\mathbb{Z}})  $ the homology groups of $W$ and $ \\hat{X}$, \nand then the image\nin the homology groups of\n$ H_*(Z', \\ensuremath{\\mathbb{Z}}) = H_*(Z, \\ensuremath{\\mathbb{Z}}) =H_*(\\Ga , \\ensuremath{\\mathbb{Z}})  $ is the same.\n\nWe apply the above consideration to the fundamental classes of the oriented manifolds $W$ and $ \\hat{X}$,\nwhich are generators of the infinite cyclic top degree homology groups $H_{2n}(W, \\ensuremath{\\mathbb{Z}})$,\nrespectively $H_{2n}( \\hat{X}, \\ensuremath{\\mathbb{Z}})$.\n\nThis implies a fortiori that $ f \\colon W \\ensuremath{\\rightarrow} Z'$ is generically finite:\nsince then the homology class  $f_* ([W])$ (which we identify to a cohomology class  by virtue of Poincar\\'e duality)\nequals  the class of $ \\hat{X}$, up to sign.\n\n\\hspace*{\\fill}$q.e.d.$\n\n\n\n{\\bf Step 3}\n\nWe claim that  $ f \\colon W \\ensuremath{\\rightarrow} Z'$ is generically 1-1 onto its image $W'$.\n\nLet in fact  $d$ be the degree of $ f \\colon W \\ensuremath{\\rightarrow} W'$.\n\nThen   $f_* ([W]) = d [W']$, hence if the class of \n$\\hat{X}$ is indivisible, then obviously $d=1$.\n\nOtherwise, observe that the divisor $W'$ is an effective $G$-invariant divisor and use our assumption.\n\n\n{\\bf Step 4}\n\nWe  established that  $f$ is birational onto its image $W'$, hence it is a \ndesingularization of $W'$.\n\n\nWe use now adjunction. We claim that, since $K_W$ is nef, there exists an effective \ndivisor $\\mathfrak A$,\ncalled adjunction divisor, such that\n$$K_W = f^* (K_{Z'} + W') - \\mathfrak A.$$\n\nThis can be shown by taking the Stein factorization\n$$ W \\ensuremath{\\rightarrow} W^N \\ensuremath{\\rightarrow} W', $$\nwhere $W^N$ is the normalization of $W'$.\n\nLet $\\sC $ be the conductor ideal  $\\mathcal{H}om (p_*\\ensuremath{\\mathcal{O}}_{W^N}, \\ensuremath{\\mathcal{O}}_{W'} )$\nviewed as an ideal $\\sC \\subset \\ensuremath{\\mathcal{O}}_{W^N}$; then the Zariski  canonical divisor \nof $W^N$ satisfies \n$$K_{W^N} =   p^*   (K_{W'}) - C =   p^*   (K_{Z'} + W') - C$$\nwhere $C$ is the Weil divisor associated to the conductor ideal (the equality on the Gorenstein locus\nof $W^N$ is shown for instance in \\cite{bucharest}, then it suffices to take the direct image from the open set to\nthe whole of $W^N$).\n\nIn turn, we would have in general \n$K_W = h^* (K_{W^N } ) - \\mathfrak B $, with $\\mathfrak B $ not necessarily effective; but, by Lemma 2.5 of \\cite{alessio},\nsee also Lemma 3.39 of \\cite{KollarMori},\nand since $- \\mathfrak B $ is h-nef, we conclude that $\\mathfrak B $ is effective.\nWe establish the claim by seting $\\mathfrak A : = \\mathfrak B + h^* C$.\n\n\nObserve that, under the isomorphism of homology groups, $f^* (K_{Z'} + W')$\ncorresponds to $(K_{Z} + \\hat{X})|_{\\hat{X}}= K_{\\hat{X}} $, in \nparticular we have\n$$ K_{\\hat{X}}^n = f^* (K_{Z'} + W')^n = (K_W + \\mathfrak A)^n . $$\nIf we assume that $K_W$ is ample, then $ (K_W + \\mathfrak A)^n \\geq (K_W)^n $,\nequality holding if and only if $  \\mathfrak A = 0$.\n\nUnder assumption (**), it follows that  $$K_{\\hat{X}}^n = |G| K_X^n = \n|G| K_{X'}^n= K_W^n,$$\nhence  $  \\mathfrak A = 0$. Since however $K_W$ is ample, it follows \nthat $f$ is an embedding.\n\nIf instead we assume that $K_W$ has the same class as $ f^* (K_{Z'} + W')$,\nwe conclude first that necessarily $\\mathfrak B =0$, and then we get that $C= 0$.\n\nHence $W'$ is normal and $W$ has canonical singularities.\n\n{\\bf Step 4}\n\nFinally, the group $G$ acts on $W$, preserving the direct summands of \nits fundamental group\n$\\Ga$. Hence, $G$ acts on the curve-factors, and the locally symmetric factors.\n\nBy assumption, moreover, it sends the summand $\\Lambda_i$ to itself, \nhence we get a well defined linear action  on each\nAbelian variety $A'_i$, so that we have a diagonal linear action of \n$G$ on $A'$.\n\nSince however the image of $W$ generates $A'$, we can extend the \naction of $G$ on $W$ to a compatible  affine\naction on $A'$.\n\nOur final step consists in showing that \n\n\\begin{lemma}\\label{affine}\nGiven a  diagonal special Inoue type manifold, the real affine type of the action of $G$ on the\nAbelian variety $ A = (A_1 \\times \\dots \\times A_r)$ is determined by the fundamental group\nexact sequence\n\n$$ 1 \\ensuremath{\\rightarrow} \\Ga= \\Pi_{i=1}^r (\\Lambda_i) \\times   \\Pi_{j=1}^h (\\pi_{g_j}) \n\\times \\Pi_{l=1}^s\n(\\pi_1 (M_l))\n\\ensuremath{\\rightarrow} \\pi_1 ( X) \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1.$$ \n\\end{lemma}\n{\\em Proof.}\nDefine as before $\\Lambda : = \\Pi_{i=1}^r (\\Lambda_i) = \\pi_1 (A)$; moreover, since all the summands in\nthe left hand side are normal in $ \\pi_1 ( X)$, set \n$$\\overline{G} : =  \\pi_1 ( X) \/ ( \\Pi_{j=1}^h (\\pi_{g_j}) \n\\times \\Pi_{l=1}^s\n(\\pi_1 (M_l)) ).$$ \n\nObserve now that $ X$ is the quotient of its universal covering \n$$\\tilde{X} = \\ensuremath{\\mathbb{C}}^m \\times \\prod_{j=1}^h\\mathbb{H}_j \\times \\prod_{l=1}^s\\mathcal{D}_l$$ by its fundamental group,\nacting diagonally (here $\\mathbb{H}_j$ is a copy of Poincar\\'e's upper half plane,while $ \\mathcal{D}_l$ is an irreducible  bounded symmetric domain\nof dimension at least two), hence we  obtain that $\\overline{G}$ acts on $\\ensuremath{\\mathbb{C}}^m$ as a group of affine transformations.\n\nLet $\\sK$ be the kernel of the associated homomorphism $$ \\alpha : \\overline{G} \\twoheadrightarrow \\im (\\alpha) =: \\hat{G} \\subset \\Aff (m, \\ensuremath{\\mathbb{C}}),$$\nand let $$\\overline{G}_1 : = \\ker (  \\alpha_L : \\overline{G} \\ensuremath{\\rightarrow} \\GL  (m, \\ensuremath{\\mathbb{C}})).$$\n\n$\\overline{G}_1 $ is obviously Abelian, and contains $\\Lambda$, and maps onto a lattice $\\Lambda' \\subset \\hat{G}$.\n\nSince $\\Lambda$ injects into $\\Lambda'$, $\\Lambda \\cap \\sK = 0 $, whence $\\sK$ injects into $G$, therefore $\\sK$ is a torsion subgroup; since $\\Lambda'$ is free, we obtain\n$$\\overline{G}_1 = \\Lambda'  \\oplus \\sK,$$\nand we finally get $$  \\sK = \\Tors (\\overline{G}_1 ), \\ \\  \\hat{G} =  \\overline{G}  \/ \\Tors (\\overline{G}_1 ).$$\n\nSince our action is diagonal, we can write $\\Lambda'  = \\oplus_{i=1}^r (\\Lambda'_i) $, and the linear action of\nthe group $G_2 : = G \/ \\sK$ preserves the summands. \n\nSince $ \\hat{G} \\subset \\Aff (\\Lambda')$, we see that $$ \\hat{G} = (\\Lambda')  \\rtimes G'_2 ,$$\nwhere $G'_2 $ is the isomorphic image of $G_2$ inside $ \\GL ( \\Lambda'))$. \n\nThis shows that the affine group  $ \\hat{G}$ is uniquely determined.\n\nFinally, using the image groups $G_{2,i}$ of $G_2$ inside $ \\GL ( \\Lambda'_i))$,\nwe can  define uniquely groups of affine transformations of $A_i$ which fully determine\nthe diagonal action of $G$ on $A$ (up to real affine automorphisms of each $A_i$).\n\n\n\\hspace*{\\fill}$q.e.d.$\n\n\n\\QED {\\em  for Theorem \\ref{special diagonal}}\n\\bigskip\n\n\nIn order to obtain weak rigidity results, one has to \nintroduce  a further invariant, called Hodge type, according to\nthe following\ndefinition. We shall return to this question  in a  sequel to this paper.\n\n\n\\begin{defin}\n   Let $X$ be an  Inoue-type manifold (or variety) of special diagonal \ntype, with\n\n$ Z = (A_1 \\times \\dots \\times A_r) \\times  (C_1 \\times \\dots \\times\nC_h) \\times (M_1 \\times \\dots \\times\nM_s)$\n\nThen an invariant of the integral representation $ G \\ensuremath{\\rightarrow} Aut (\\Lambda_i)$\nis its {\\bf Hodge type},\nwhich is the datum, given the decomposition of $\\Lambda_i \\otimes \\ensuremath{\\mathbb{C}}$\nas the sum of\nisotypical components $$\\Lambda_i \\otimes \\ensuremath{\\mathbb{C}} = \\oplus_{\\chi \\in \\Irr (G)}\nU_{i, \\chi}$$\nof the dimensions   $$\\nu (i, \\chi): = dim_{\\ensuremath{\\mathbb{C}}} U_{i, \\chi}  \\cap\nH^{1,0}(A_i) $$\nof the Hodge summands for non real representations.\n\\end{defin}\n\n\n\n\n\n\n\\section {Genus $5$ curves having a $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4$-action}\\label{section2}\n\nThe following is well known (see however section 1 of \\cite{burniat1}).\n\n\\begin{lemma}\\label{genus1}\nLet $E_1$ be a compact curve of genus $1$  and assume $ G_1: = (\\ensuremath{\\mathbb{Z}}\/ 2\n\\ensuremath{\\mathbb{Z}})^n \\subset Aut (E_1)$. Then $ n \\leq 3$, and, for $n=3$,\n$ E_1\/ G_1 \\cong \\ensuremath{\\mathbb{P}}^1$ with quotient map  branched on exactly\n$4$ points $P_1, \\dots, P_4$.\nThe covering  $ E_1 \\ensuremath{\\rightarrow}  E_1\/ G_1$ factors through multiplication by \n2 in $E_1$.\n\\end{lemma}\n\n\n\\begin{lemma}\\label{genus5}\nLet $D$ be a compact curve of genus $5$  and assume $ H : = (\\ensuremath{\\mathbb{Z}}\/ 2\n\\ensuremath{\\mathbb{Z}})^n \\subset Aut (D)$.\nThen $ n \\leq 4$, and, if $ n = 4$, $ D \/ H \\cong \\ensuremath{\\mathbb{P}}^1$ and the quotient map is branched on exactly\n$5$ points $P_1, \\dots, P_5$.\n\\end{lemma}\n\\proof\nBy the Hurwitz' formula, one has, setting $ D \/ H = C$, and setting\n$h$= genus ($C$),\n$$ 8 = 2^n (2h-2 + \\frac{m}{2}) \\Leftrightarrow  2^{n-4} (4h-4 + m) = 1$$\nwhere $m$ is the number of branch points $P_1, \\dots, P_m$. Hence\n$n \\leq 4 $. If $n=4$, then $h=1$ is not possible,\nsince in this case  the  abelianization of  $\\pi_1 (C \\setminus {P_1}) $ would equal $\\pi_1\n(C)$, and one would have $m=0$,\na contradiction.\nHence $h=0$ and $m=5$.\n\n\\hspace*{\\fill}$q.e.d.$\n\nThe following geometrical game is based on the fact that the 15\nintermediate double covers of\n   $ D \/ H = \\ensuremath{\\mathbb{P}}^1$ are 5 elliptic curves (branched each on 4 of the 5\nbranch points) and 10 rational curves\n(branched each on 2 of the 5 branch points).\nLet $A_i$ be the elliptic curve branched on all the five points with\nexclusion of $P_i$: then\n$D \\ensuremath{\\rightarrow} D \/ H $ factors as $$D \\ensuremath{\\rightarrow}  A_i \\ensuremath{\\rightarrow} A_i \\ensuremath{\\rightarrow} D \/ H $$\nwhere the middle map is multiplication by 2, and $D \\ensuremath{\\rightarrow}  A_i$ is the\nquotient by an involution with\n   fixed points; the number of  fixed points is exactly 8, since, if $g\n\\in H$, the fixed set $\\Fix(g)$ is an $H$-orbit, and has\ntherefore cardinality equal to a multiple of $8$. The other 10\ninvolutions have no fixed points, hence they\nyield each an unramified covering of a curve $C_j$ of genus $3$.\n\nWe try now to stick to Inoue's original notation, except that we refuse to\nuse the classical symbol for  the {\\em Weierstrass $\\wp$-function} to denote\nthe Legendre function $\\sL$;   $\\sL$ is a  homographic transform of\n  the Weierstrass function, but not equal to the Weierstrass function.\n\n  The Legendre function satisfies the quadratic relation (see \\cite{burniat1})\n  $$ y^2 = (\\sL^2 - 1) ( \\sL^2 - a^2) .$$\n\nLet $E_1$, $E_2$ be two complex elliptic curves. We assume $E_i = \\ensuremath{\\mathbb{C}}\n\/ \\langle 1, \\tau_i \\rangle$. Moreover, we denote by\n$z_i$ a uniformizing parameter on $E_i$.\n\n  Then $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3 =\n\\langle \\gamma_1, \\gamma_2, \\gamma_3 \\rangle$ acts on\n$E_i$ by\n\\begin{itemize}\n   \\item[-] $\\gamma_1(z_i) = -z_i$;\n\\item[-] $\\gamma_2(z_i) = z_i + \\frac 12$;\n\\item[-] $\\gamma_3(z_i) = z_i + \\frac{\\tau_i}{2}$.\n\\end{itemize}\n\n We consider the Legendre $\\sL$-function for $E_i$ and denote it by \n$\\sL_i$,  for $i=1,2$:\n$\\sL_i$ is a meromorphic\nfunction on $E_i$ and $\\sL_i \\colon E_i \\ensuremath{\\rightarrow} \\ensuremath{\\mathbb{P}}^1$ is a double cover \nramified in\n$\\pm 1, \\pm a_i \\in \\ensuremath{\\mathbb{P}}^1 \\setminus \\{0, \\infty\\}$.\n\nIt is well known that we have (cf. \\cite{inoue}, lemma 3-2, and also \ncf. \\cite{burniat1}, pages 52-54, section 1 for an algebraic \ntreatment):\n\\begin{itemize}\n   \\item[-] $\\sL_i(\\frac 12) = -1$, $\\sL_i(0) = 1$,\n$\\sL_i(\\frac{\\tau_i}{2}) = a_i$,  $\\sL_i(\\frac{1+\\tau_i}{2}) = - \na_i$;\n\\item[-] let $b_i:= \\sL_i(\\frac{\\tau_i}{4})$: then $b_i^2 = \na_i$;\n\\item[-] $\\frac{\\rm{d} \\sL_i}{\\rm{d} z_i} (z_i) = 0$ if and \nonly if \n$z_i \\in \\{0, \\frac 12,\\frac{\\tau_i}{2}, \\frac{1+\\tau_i}{2} \n\\}$, since $\\frac { d x_i}{ \\sL_i} = dz_i$.\n\\end{itemize}\n\nMoreover \n$$\n\\sL_i(z_i) = \\sL_i(z_i+1) = \\sL_i(z_i+\\tau_i) = \\sL_i(-z_i) = \n-\\sL_i(z_i+ \\frac 12),\n$$\n$$\n\\sL_i(z_i + \\frac{\\tau_i}{2}) = \n\\frac{a_i}{\\sL_i(z_i)}.\n$$\nWe consider now the vector space \n$V_i:=H^0(E_i, \\ensuremath{\\mathcal{O}}_{E_i}(2[0]))$ \n(for $i=1,2$), and note that $V_i \n\\cong \\ensuremath{\\mathbb{C}}^2$ with basis\n$1, \\sL_i$, since \n$$div (1 - \\sL_i) = 2 [0] \n-  {\\it Poles} (\\sL_i) .$$ .\n\nObserve that $[- \\frac 14] + [\\frac \n14]$ is a $(\\ensuremath{\\mathbb{Z}}\/ 2\\ensuremath{\\mathbb{Z}})^2 =\\langle \n\\gamma_1, \\gamma_2 \\rangle$ - \ninvariant divisor, hence $V_i$\nis a $(\\ensuremath{\\mathbb{Z}} \/ 2\\ensuremath{\\mathbb{Z}})^2$ - module and \nsplits in its isotypical components as\n$$ V_i = V_i^{++} \\oplus \nV_i^{+-},\n$$ since $1$ is invariant under $(\\ensuremath{\\mathbb{Z}}\/2\\ensuremath{\\mathbb{Z}})^2$ and $\\sL_i$ \nis invariant under $\\gamma_1$ and is an eigenvector with\neigenvalue \n$-1$ of $\\gamma_2$.\n\n\nIf $c \\in \\ensuremath{\\mathbb{C}} \\setminus \\{ \\pm 1, \\pm a_i, \\pm \na_1a_2 \\}$, then the divisor\n$$ D_c:=\\{ (z_1,z_2) \\in E_1 \\times E_2 \n\\ | \\ \\sL_1(z_1) \\sL_2(z_2) = c \\}\n$$\n\nof bidegree $(2,2)$ is a \nsmooth curve of genus $5$. More precisely,\n$$\n\\ensuremath{\\mathcal{O}}_{E_1 \\times \nE_2}(D_c) \\cong p_1^* \\ensuremath{\\mathcal{O}}_{E_1}(2[0]) \\otimes p_2^* \n\\ensuremath{\\mathcal{O}}_{E_2}(2[0]).\n$$\n\nConsider the product action of $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3 \n\\times (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3$ \non $E_1 \\times E_2$.\n\\begin{rem} 1) It is easy \nto see that $D_c$ is invariant under the subgroup $H \n\\leq (\\ensuremath{\\mathbb{Z}}\/2 \n\\ensuremath{\\mathbb{Z}})^3 \\times (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3$ given by\n$$ (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3 \\cong H:= \n\\langle (\\begin{pmatrix} 1\\\\ 0 \\\\ 0 \n\\end{pmatrix},\\begin{pmatrix} \n0\\\\0\\\\0\n\\end{pmatrix}),(\\begin{pmatrix} 0\\\\ 0 \\\\ 0 \n\n\\end{pmatrix},\\begin{pmatrix} 1\\\\0\\\\0 \n\\end{pmatrix}),(\\begin{pmatrix} \n0\\\\ 1 \\\\ \n0\n\\end{pmatrix},\\begin{pmatrix} 0\\\\1\\\\0 \\end{pmatrix}) \n\\rangle,\n$$\n\nwhere the coordinates are taken with respect to the \nbasis $\\gamma_1, \n\\gamma_2, \\gamma_3$ on each factor.\n\n2) Moreover, \nif we choose $c:= b_1b_2$, then we see that for \n$(z_1,z_2) \\in \nD_{b_1b_2}$:\n$$\n\\sL_1(z_1 + \\frac{\\tau_1}{2})\\sL_2(z_2 + \n\\frac{\\tau_2}{2}) = \n\\frac{a_1a_2}{\\sL_1(z_1)\\sL_2(z_2)} \n=\n\\frac{a_1a_2}{b_1b_2} = b_1b_2,\n$$ whence $D_{b_1b_2}$ is invariant \nunder\n$$ (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^4 \\cong G:= H \\oplus \\langle (\\begin{pmatrix} \n0\\\\ 0 \n\\\\ 1 \\end{pmatrix},\\begin{pmatrix} 0\\\\0\\\\1\n\\end{pmatrix}) \\rangle \\leq (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3 \\times (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3.\n$$\n\\end{rem}\n\nWe want to show that the converse holds. More precisely, we prove the following\n\\begin{prop}\\label{genus5,2}\n   Let $f \\colon D \\ensuremath{\\rightarrow} \\ensuremath{\\mathbb{P}}^1$ be the maximal $G:= (\\ensuremath{\\mathbb{Z}}\/2\n\\ensuremath{\\mathbb{Z}})^4$-covering branched in $5$ given points $p_1, \\ldots , p_5 \\in\n\\ensuremath{\\mathbb{P}}^1$. Then there are two elliptic curves $E_1$, $E_2$ such that $D\n\\subset E_1 \\times E_2$ is a $G$-invariant divisor with\n$$\n\\ensuremath{\\mathcal{O}}_{E_1 \\times E_2}(D) \\cong p_1^* \\ensuremath{\\mathcal{O}}_{E_1}(2[0]) \\otimes p_2^*\n\\ensuremath{\\mathcal{O}}_{E_2}(2[0]).\n$$ Choosing appropriate coordinates we can moreover assume that $D=\n\\{ (z_1,z_2) \\in E_1 \\times E_2 \\ | \\ \\sL_1(z_1)\n\\sL_2(z_2) = b_1b_2 \\}$.\n\\end{prop}\n\\begin{proof}\n   Let $e_1, e_2,e_3,e_4$ be a basis of the $\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}}$ - vectorspace\n$G$ and let $D \\rightarrow \\ensuremath{\\mathbb{P}}^1$ branched in $p_1, \\ldots\np_5$ be given by the {\\em appropriate orbifold homomorphism}\n$$\n\\varphi : \\mathbb{T}(2,2,2,2,2) := \\langle x_1, \\ldots , x_5 |\n\\prod_{i=1}^5 x_i, x_1^2, \\ldots, x_5^2 \\rangle \\rightarrow\n(\\ensuremath{\\mathbb{Z}}\/2\\ensuremath{\\mathbb{Z}})^4,\n$$ where $\\varphi(x_i) = e_i$ for $1\\leq i \\leq4$, $\\varphi(x_5) =\ne_5:=e_1 +e_2+e_3+e_4$. Then Hurwitz' formula shows that\n$D$ is a smooth curve of genus $5$.\n\nNote that the only elements of $G$ having fixed points on $D$ are the 5 elements \n$e_i$ ($1 \\leq i \\leq 5$), and each of them has exactly $8$\nfixed points on $D$. Hence, $E_i:=D\/\\langle e_i\\rangle$ is an\nelliptic curve.\n\nWe get therefore $5$ elliptic curves (as intermediate covers of $D\n\\rightarrow \\ensuremath{\\mathbb{P}}^1$), all endowed with a $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3$ - action.\n\nChoose two of these elliptic curves, say $E_1, E_2$, and consider the morphism\n$$ i \\colon D \\rightarrow D\/\\langle e_1\\rangle \\times D\/\\langle\ne_2\\rangle = E_1 \\times E_2.\n$$\n\nThen $D \\cdot E_i =2$ and $i$ is an embedding of $D$ as a $(\\ensuremath{\\mathbb{Z}} \/ 2\n\\ensuremath{\\mathbb{Z}})^4$ - invariant divisor of bidegree $2$.\n\nWe fix the origin in both elliptic curves so that $D= \\{s=0 \\}$, where $s \\in H^0(E_1 \\times E_2,\np_1^*(\\ensuremath{\\mathcal{O}}_{E_1}(2[0]))) \\otimes\np_2^*(\\ensuremath{\\mathcal{O}}_{E_2}(2[0])))^G$.\n\nIt remains to show that we can assume $D$ to be of the form $\\{\n(z_1,z_2) \\in E_1 \\times E_2 \\ | \\ \\sL_1(z_1) \\sL_2(z_2) =\nb_1b_2 \\}$.\n\nFor this we show that\n$$H^0(E_1 \\times E_2, p_1^*(\\ensuremath{\\mathcal{O}}_{E_1}(2[0]))) \\otimes\np_2^*(\\ensuremath{\\mathcal{O}}_{E_2}(2[0])))^H \\cong \\ensuremath{\\mathbb{C}}^2.$$ In fact,\n\n\\begin{multline*}\n   H^0(E_1 \\times E_2, p_1^*(\\ensuremath{\\mathcal{O}}_{E_1}(2[0]))) \\otimes\np_2^*(\\ensuremath{\\mathcal{O}}_{E_2}(2[0]))) = V_1 \\otimes V_2 = \\\\ = (V_1^{+++} \\oplus\nV_1^{++-}) \\otimes (V_2^{+++} \\oplus V_2^{++-}),\n\\end{multline*} whence\n\\begin{multline*} H^0(E_1 \\times E_2, p_1^*(\\ensuremath{\\mathcal{O}}_{E_1}(2[0])))\n\\otimes p_2^*(\\ensuremath{\\mathcal{O}}_{E_2}(2[0])))^H = \\\\ = (V_1^{+++}\n\\otimes V_2^{+++}) \\oplus (V_1^{++-} \\otimes V_2^{++-}).\n\\end{multline*}\n\nTherefore we have a pencil of  $H$-invariant divisors $D_c:=\\{\n(z_1,z_2) \\in E_1 \\times E_2 \\ | \\ \\sL_1(z_1) \\sL_2(z_2) = c\n\\}$. It is now obvious that $D_c$ is $G$ - invariant iff $c = \\pm b_1b_2$.\nThe change of sign for $b_i$ is achieved by changing the point $\\frac{\\tau_i}{4}$\nwith $\\frac{\\tau_i}{4} + \\frac{1}{2}$.\n\\end{proof}\n\nConsider now for the moment  the action of $(\\ensuremath{\\mathbb{Z}} \/\n2\\ensuremath{\\mathbb{Z}})^2 = \\langle \\gamma_1, \\gamma_2\\rangle$ on $E_1$ and $E_2$, and\nthe induced product action on $E_1 \\times E_2$. Assume $H \\cong (\\ensuremath{\\mathbb{Z}}\n\/ 2 \\ensuremath{\\mathbb{Z}})^3 \\leq (\\ensuremath{\\mathbb{Z}} \/ 2\\ensuremath{\\mathbb{Z}})^2 \\times (\\ensuremath{\\mathbb{Z}} \/ 2\\ensuremath{\\mathbb{Z}})^2$,\nsuch that\n\\begin{enumerate}\n  \\item on each factor $E_i$ it induces the action \nof $(\\ensuremath{\\mathbb{Z}} \/ 2\\ensuremath{\\mathbb{Z}})^2= \n\\langle \\gamma_1, \\gamma_2 \\rangle$;\n\\item there \nis an $H$-invariant pencil in $\\ensuremath{\\mathbb{P}}(V_1 \\otimes V_2)$, \ni.e., $\\dim \n(V_1 \\otimes V_2)^H = 2$.\n\\end{enumerate}\n\nThen it is easy to see \nthat we can choose $(\\gamma_1, 0)$, \n$(0,\\gamma_1)$, $(\\gamma_2, \n\\gamma_2)$ as basis of $H$.\n\nTherefore\n$$ (V_1 \\otimes V_2)^{+++} = \n(V_1^{+++} \\otimes V_2^{+++}) \\oplus \n(V_1^{++-} \\otimes \nV_2^{++-}),\n$$ and the $H$- invariant pencil is given by $D_c = \n\\{(z_1,z_2) \\in \nE_1 \\times E_2 \\ | \\ \\sL_1(z_1) \\sL_2(z_2) = c \n\\}$.\n\nConsider now $G:= H \\oplus \\langle (\\gamma_3, \\gamma_3) \\rangle \n\\cong \n(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4$. \n\nThen $D:= \\{(z_1,z_2) \\in E_1\n\\times E_2 \\ | \\ \n\\sL_1(z_1) \\sL_2(z_2) = b_1b_2\\}$.\n\n\n\n\n\\begin{rem} \\\n\\begin{enumerate}\n \n\\item The restriction map  $$ H^0(E_1 \\times E_2, p_1^*\\ensuremath{\\mathcal{O}}_{E_1}(2[0]) \\otimes p_2^* \n\\ensuremath{\\mathcal{O}}_{E_2}) \\rightarrow H^0(D, p_1^*\\ensuremath{\\mathcal{O}}_{E_1}(2[0]) \\otimes\np_2^* \n\\ensuremath{\\mathcal{O}}_{E_2} | D)\n$$ is an isomorphism of $H$-modules.\n\\item There is a \npencil of such $H$- invariant divisors of degree 4 \non $D$. But there \nis no one which is invariant under $(\\ensuremath{\\mathbb{Z}} \/2\n\\ensuremath{\\mathbb{Z}})^4$.\n\\end{enumerate}\n\\end{rem}\n\n\\begin{proof}[Proof \nof rem. ( 1).] Let be $S:= E_1 \\times E_2$ and for \nsimplicity we \nwrite\n$$\n\\ensuremath{\\mathcal{O}}_S(2,0) := p_1^*\\ensuremath{\\mathcal{O}}_{E_1}(2[0]) \\otimes p_2^* \n\\ensuremath{\\mathcal{O}}_{E_2}.\n$$ Then we consider the exact sequence:\n$$ 0 \\rightarrow \n\\ensuremath{\\mathcal{O}}_S(-D) \\otimes  \\ensuremath{\\mathcal{O}}_S(2,0) \\rightarrow \n\\ensuremath{\\mathcal{O}}_S(2,0) \\rightarrow \n\\ensuremath{\\mathcal{O}}_S(2,0) \\otimes \\ensuremath{\\mathcal{O}}_D \\rightarrow 0.\n$$ By K\\\"unneth's formula \nwe get\n\\begin{itemize}\n  \\item[i)] $h^0(\\ensuremath{\\mathcal{O}}_S(-D) \\otimes \n\\ensuremath{\\mathcal{O}}_S(2,0)) = \nh^0(\\ensuremath{\\mathcal{O}}_{E_1})h^0(\\ensuremath{\\mathcal{O}}_{E_2}(-2) = 0$;\n\\item[ii)] \n$h^1(\\ensuremath{\\mathcal{O}}_S(-D) \\otimes  \\ensuremath{\\mathcal{O}}_S(2,0)) = 2$.\n\\end{itemize} \nTherefore\n$$ r \\colon H^0(S, \\ensuremath{\\mathcal{O}}_S(2,0)) \\cong \\ensuremath{\\mathbb{C}}^2 \\rightarrow \nH^0(S, \n\\ensuremath{\\mathcal{O}}_S(2,0) \\otimes \\ensuremath{\\mathcal{O}}_D)\n$$ is injective. Since $D$ is not \nhyperelliptic it follows by \nClifford's theorem that $h^0(S, \n\\ensuremath{\\mathcal{O}}_S(2,0) \\otimes \\ensuremath{\\mathcal{O}}_D) \\leq 2$.\nThis implies that $h^0(S, \n\\ensuremath{\\mathcal{O}}_S(2,0) \\otimes \\ensuremath{\\mathcal{O}}_D) = 2$ and $r$ is \nan isomorphism (of \n$H$-modules).\n\\end{proof}\n\n\\begin{rem}\n  This implies that\n$$ H^0(S, \n\\ensuremath{\\mathcal{O}}_S(2,0) \\otimes \\ensuremath{\\mathcal{O}}_D) \\cong \\ensuremath{\\mathbb{C}}^2 = V^{+++} \\oplus \nV^{++-}.\n$$\n\\end{rem}\n\n\n\\section{Inoue surfaces with $p_g=0$ and \n$K_S^2 = 7$}\n\nIn \\cite{inoue} the author describes the construction \nof a family of \nminimal surfaces of general type $S$ with $p_g= 0$, \n$K_S^2\n= 7$.\n\nWe briefly recall the construction of these surfaces \nand, for lack of \nreference, we calculate $K_S^2$ and $p_g(S)$.\n\nFor \n$i \\in \\{1,2,3,4\\}$, let $E_i := \\ensuremath{\\mathbb{C}} \/ \\langle 1, \\tau_i \\rangle$ \nbe \na complex elliptic curve. Denoting again by $z_i$ a\nuniformizing \nparameter of $E_i$, we consider the following five \ninvolutions on \n$T:=E_1 \\times E_2\\times E_3 \\times E_4$:\n\\begin{itemize}\n\\item[] $ g_1(z_1,z_2,z_3,z_4) = \n(-z_1+\\frac 12, z_2 + \\frac 12, z_3,z_4)$,\n\\item[] $g_2(z_1,z_2,z_3,z_4) \n= (z_1, -z_2 + \\frac 12, z_3 +\\frac 12,-z_4+ \\frac 12)$,\n\\item[] $g_3(z_1,z_2,z_3,z_4) = (z_1+\\frac 12, z_2, -z_3 + \\frac 12,-z_4  + \n\\frac 12)$,\n\\item[]  $g_4(z_1,z_2,z_3,z_4) = (z_1, z_2, -z_3,-z_4)$,\n\\item[] $g_5(z_1,z_2,z_3,z_4) = (z_1+\\frac{\\tau_1}{2}, \nz_2+\\frac{\\tau_2}{2}, \nz_3+\\frac{\\tau_3}{2},z_4+\\frac{\\tau_4}{2})$.\n\\end{itemize}\n Then $G:= \\langle g_1,g_2,g_3,g_4,g_5 \\rangle \\cong (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^5$.\n\nConsider\n$$\n\\hat{X} := \\{ (z_1,z_2,z_3,z_4)) \\in T \\ | \\ \\sL_1(z_1)\n\\sL_2(z_2)\\sL_3(z_3)  = b_1b_2b_3, \\ \\sL_3(z)\\sL_4(z_4) =\nb_3b_4\\}.\n$$ Then\n\\begin{itemize}\n   \\item[-] $\\hat{X}$ is a smooth complete intersection of two\nhypersurfaces in $T$ of respective multidegrees $(2,2,2,0)$ and\n$(0,0,2,2)$;\n\\item[-] $\\hat{X}$ is invariant under the action of $G$, and $G$ acts\nfreely on $\\hat{X}$.\n\\end{itemize}\n\n\nThe above equations show that  $\\hat{X}$ is the complete intersection of a $G$ -\ninvariant divisor $X_1$ in the linear system\n$$\n\\ensuremath{\\mathbb{P}}(H^0(E_1 \\times E_2 \\times E_3 \\times E_4, p_1^*\\ensuremath{\\mathcal{O}}_{E_1}(2[0])\n\\otimes p_2^*\\ensuremath{\\mathcal{O}}_{E_2}(2[0]) \\otimes\np_3^*\\ensuremath{\\mathcal{O}}_{E_3}(2[0]) \\otimes p_4^*\\ensuremath{\\mathcal{O}}_{E_4})),\n$$  with a $G$-invariant divisor $X_2$ ($\\cong E_1 \\times E_2 \\times\nD$), where $D$ is a curve of genus $5$, a Galois cover\nwith group $(\\ensuremath{\\mathbb{Z}}\/ 2 \\ensuremath{\\mathbb{Z}})^4$ of the projective line  ramified in 5\npoints) in the linear system\n$$\n\\ensuremath{\\mathbb{P}}(H^0(E_1 \\times E_2 \\times E_3 \\times E_4, p_1^*\\ensuremath{\\mathcal{O}}_{E_1} \\otimes\np_2^*\\ensuremath{\\mathcal{O}}_{E_2} \\otimes p_3^*\\ensuremath{\\mathcal{O}}_{E_3}(2[0])\n\\otimes p_4^*\\ensuremath{\\mathcal{O}}_{E_4}(2[0]))).\n$$\n\n\n\\begin{rem}\n   It is easy to see from the above explicit description that $G$ acts\nfreely on $\\hat{X}$. Note that the action of $G$ has fixed points\non $T$, and even on $X_2$.\n\\end{rem}\n\n\\begin{defin} A smooth projective algebraic surface $S:= \\hat{X} \/ G$ as above\nis called an {\\em Inoue surface} with $K_S^2 =7$ and\n$p_g=0$.\n\n\\end{defin} \n \\begin{rem}\n   It is immediate to see that the involution $g_4$ acts freely on  $ \\hat{X} $\n   and trivially on $(E_1 \\times E_2)$.\n   \n    It follows therefore that,\n    setting $ \\hat{X}_2 : =  \\hat{X} \/ g_4 $,\n    $S$ has another\n   representation as $ \\hat{X}_2 \/ (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^4$, where\n   $ \\hat{X}_2$ is a divisor in the product $(E_1 \\times E_2 \\times C)$,\n  and $C$ is a smooth curve of genus $3$.\n   \nThis was the representation of $S$ announced in \\cite{survey}, and it follows from our results that\nthis representation is also unique.\n\n\\end{rem}\n\n\n\nWe will show in the next lemma that $S$ is a minimal\nsurface of general type with $K_S^2 = 7$ and $p_g=0$.\n\n\n\\begin{lemma}\\label{invariants}\n  Let $S=\\hat{X} \/G$ be as above. \nThen $S$ is a minimal surface of \ngeneral type with $K_S^2 = 7$ and \n$p_g=0$.\n\\end{lemma}\n\n\\begin{proof}\n\nSince $\\hat{X}$ is of general \ntype, also $S$ is of general type being \nan \\'etale quotient of \n$\\hat{X}$ by a finite group.\n\n\nWe first remark that $\\hat{X}$ is a \n$G$-invariant hypersurface of \nmultidegree $(2,2,4)$ in $W:=E_1 \n\\times E_2 \\times D$,\nwhere $D \\subset E_3\\times E_4$ is a smooth \ncurve of genus 5 given by \nthe equation\n$$\n\\{ (z_3,z_4))\\in E_3 \n\\times E_4 \\ | \\ \\sL_3(z)\\sL_4(z_4) = b_3b_4\\}.\n$$\n\nBy the adjunction \nformula, the canonical divisor of $\\hat{X}$ is the \nrestriction to \n$\\hat{X}$ of a divisor of multidegree $(2,2,12)$\non $W$.\n\nTherefore \nwe can calculate (denoting by $F_i$ the fibre of the \nprojection of \n$W$ on the $(j,k)$-th coordinate (with $\\{i,j,k\\} \n=\n\\{1,2,3\\}$).\n\\begin{multline*} K_{\\hat{X}}^2 = ((K_W + \n[\\hat{X}])|S)^2 = \n(2F_1+2F_2+12F_3)^2(2F_1+2F_2+4F_3)=\\\\ = \n(8F_1F_2+\n48F_1F_3 + 48 F_2F_3)(2F_1+2F_2+4F_3)= \\\\ = 32F_1F_2F_3 + \n96F_1F_2F_3+96F_1F_2F_3 = 224 = 7 \\cdot 2^5.\n\\end{multline*}\n\nSince $G$ acts freely on $\\hat{X}$, we obtain\n$$ \nK_S^2 = \\frac{224}{|G|} = \\frac{224}{2^5} = 7.\n$$\n\nMoreover,  \nconsider the exact sequence\n\\begin{equation}\\label{adjunction}\n  0 \n\\rightarrow \\ensuremath{\\mathcal{O}}_W(K_W) \\rightarrow \\ensuremath{\\mathcal{O}}_W(K_W + [\\hat{X}]) \n\\rightarrow \\omega_{\\hat{X}} \\rightarrow 0.\n\\end{equation}\n\nUsing \nK\\\"unneth's formula and Kodaira's vanishing theorem, we \nget:\n\\begin{itemize}\n\\item[-] $\\dim H^0(W, \\ensuremath{\\mathcal{O}}_W(K_W)) = \n5$,\n\\item[-] $\\dim H^0(W, \\ensuremath{\\mathcal{O}}_W(K_W+ [X])) = 32$,\n\\item[-] $H^i(W, \n\\ensuremath{\\mathcal{O}}_W(K_W+ [X])) = 0$, for $i=1,2,3$,\n\\item[-] $\\dim H^1(W, \n\\ensuremath{\\mathcal{O}}_W(K_W)) = 1 +5+5 = 11$,\n\\item[-] $\\dim H^2(W, \\ensuremath{\\mathcal{O}}_W(K_W)) = \nq(W) (= q(\\hat{X})) = 7$.\n\\end{itemize} Therefore, by the long exact \nsequence associated to \n(\\ref{adjunction}) we get:\n$$ p_g(\\hat{X}) = \nh^0(\\hat{X}, \\omega_{\\hat{X}}) = 32+11-5 = 38.\n$$ \nTherefore\n$$\n\\chi(\\ensuremath{\\mathcal{O}}_{\\hat{X}}) = 1+p_g(\\hat{X}) -q(\\hat{X}) = 1 + \n38 -7 =32.\n$$\n\nThis implies that $\\chi(\\ensuremath{\\mathcal{O}}_S) = 1$. In order to show \nthat $p_g(S) = \n0$, it suffices to show that $q(S) =0$. Using the \nfact that\n$$ i^* \\colon H^0(W, \\Omega^1_W) \\rightarrow H^0(\\hat{X}, \n\\Omega^1_{\\hat{X}})\n$$ is an isomorphism and that\n$$ H^0(W, \n\\Omega^1_W) = H^0(E_1, \\Omega^1_{E_1})\\oplus H^0(E_2, \n\\Omega^1_{E_2}) \\oplus H^0(D, \\Omega^1_D),\n$$ it is easy to see that \n$H^0(W, \\Omega^1_W)^G = H^0(S, \n\\Omega^1_S)=0$.\n\\end{proof}\n\n\\subsection{The torsion group of Inoue surfaces with $K_S^2 =7$}\nThe aim of the section is to prove the following\n\\begin{theo}\\label{homology}\nLet $S$ be an Inoue surface with $K_S^2 = 7$. Then \n$$\nH_1(S, \\ensuremath{\\mathbb{Z}}) \\cong \\ensuremath{\\mathbb{Z}}\/4\\ensuremath{\\mathbb{Z}} \\times (\\ensuremath{\\mathbb{Z}} \/2 \\ensuremath{\\mathbb{Z}})^4.\n$$\n\\end{theo}\nIt is clear from the construction that the fundamental group of an Inoue surface sits in an exact sequence\n\\begin{equation}\\label{fundex}\n1 \\rightarrow \\mathbb{Z}^4 \\times \\pi_5 \\rightarrow \\pi_1(S)\n\\rightarrow G \\cong (\\mathbb{Z}\/2 \\mathbb{Z})^5\n\\rightarrow 1,\n\\end{equation}\n where $\\pi_5$ denotes the fundamental group of a compact curve\n of genus five.\n \n Observe that after moding out by $\\ensuremath{\\mathbb{Z}}^4$ in the exact sequence (\\ref{fundex}) we obtain the orbifold exact sequence (plus a summand $\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}}$) of the maximal $(\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^4$-covering of $\\ensuremath{\\mathbb{P}}^1$ ramified in 5 points:\n \\begin{equation}\\label{orbifold}\n 1 \\ensuremath{\\rightarrow} \\pi_5 \\ensuremath{\\rightarrow} \\pi_1^{orb} \\times \\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}}  \\ensuremath{\\rightarrow} G \\cong (\\mathbb{Z}\/2 \\mathbb{Z})^4 \\times \\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}} \\ensuremath{\\rightarrow} 1,\n \\end{equation}\n where $\\pi_1^{orb}:=\\pi_1^{orb}(\\ensuremath{\\mathbb{P}}^1 \\setminus\\{p_1, \\ldots , p_5\\};2,2,2,2,2)$.\n \n The proof of the above theorem will be divided in several steps, and we shall first \n prove  some auxiliary results.\n\\begin{lemma}\\label{action}\nAssume that there is an exact sequence of groups\n$$\n1 \\ensuremath{\\rightarrow} \\Lambda \\ensuremath{\\rightarrow} \\Gamma' \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n$$\nwhere $\\Lambda$ and $G$ are abelian. \n\nAssume moreover that\n\\begin{itemize}\n\\item[($\\#$)]  $\\Lambda$ admits a a system $L$ of generators with the following property:\n $ \\forall h \\in L$  $ \\exists g \\in G$ such that \n$$\nghg^{-1} = -h.\n$$\n\\end{itemize}\nThen $2 \\Lambda \\subset [\\Gamma', \\Gamma']$ and, in particular, we have an exact sequence \n\\begin{equation}\\label{lem1}\n\\Lambda\/ 2 \\Lambda \\ensuremath{\\rightarrow} \\Gamma'^{ab} \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nLet $h \\in \\Lambda$, $g \\in G$ be such that $ghg^{-1} = -h$. Then $[g,h] = -2h$, whence $2h \\in [\\Gamma', \\Gamma']$. \nSince this holds for all $h \\in L$ and $L$ generates $\\Lambda$ the claim follows.\n\n\\end{proof}\n\n\\begin{rem}\\label{reduction}\ni) Assume that there is an exact sequence of groups\n$$\n1 \\ensuremath{\\rightarrow} H \\ensuremath{\\rightarrow} \\Gamma \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n$$\nwhere $G$ is abelian. Defining $\\Gamma' := \\Gamma \/ [H,H]$,  we have an exact sequence\n\\begin{equation}\\label{exrem}\n1 \\ensuremath{\\rightarrow} \\Lambda:= H^{ab} \\ensuremath{\\rightarrow} \\Gamma' \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n\\end{equation}\nand we have\n$$\n(\\Gamma')^{ab} = \\Gamma^{ab}.\n$$\nSuppose now that assumption ($\\#$) of lemma \\ref{action} is satisfied for the exact sequence (\\ref{exrem}). Then by lemma \\ref{action} we have an exact sequence \n\\begin{equation}\n\\tilde{\\Lambda} :=  \\Lambda\/2 \\Lambda \\ensuremath{\\rightarrow} \\Gamma^{ab} \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1.\n\\end{equation}\n\n\\noindent \nii) Define $\\Gamma'':= \\Gamma' \/ 2 \\Lambda$. Then we have an exact sequence\n$$\n1 \\ensuremath{\\rightarrow} \\tilde{\\Lambda} \\ensuremath{\\rightarrow} \\Gamma'' \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n$$\nand $(\\Gamma'')^{ab} = (\\Gamma')^{ab} = \\Gamma^{ab}$.\n\nChoose generators $(g_i)_{i \\in I}$ of $G$ and choose for each $g_i$ a lift $\\gamma_i$ to $\\Gamma''$. Moreover, let $(\\lambda_j)_{j \\in J}$ be generators of $\\Lambda$ and denote their images in $\\tilde{\\Lambda}$ by $\\tilde{\\lambda}_j$. Then obviously $\\Gamma'' = \\langle (g_i)_{i \\in I}, (\\tilde{\\lambda}_j)_{j \\in J} \\rangle$.\n\nAssume also that \n\\begin{itemize}\n\\item[($\\# \\#$)] $\\gamma_i \\lambda_j \\gamma_i^{-1} = \\pm \\lambda_j$, $\\forall i \\in I, \\ \\forall j \\in J$.\n\\end{itemize}\nThen (since $2\\tilde{\\Lambda} = 0$) we have $[\\gamma_i, \\tilde{\\lambda}_j] = 0$ (in $\\Gamma''$), $\\forall i \\in I, \\ \\forall j \\in J$.\n\n In particular this implies that \n$$\n\\Gamma'' \/ \\langle [\\gamma_i, \\gamma_j], i,j \\in I \\rangle = (\\Gamma'')^{ab} = \\Gamma^{ab}.\n$$\n\\end{rem}\n\nSince the genus 5 curve $D$ is an ample divisor $D \\subset E_3 \\times E_4$, we have by Lefschetz' theorem a surjective morphism\n $$\n\\varphi:  \\pi_5 \\cong \\pi_1(D) \\rightarrow \\pi_1(E_3 \\times E_4) \\cong \\ensuremath{\\mathbb{Z}}^4.\n $$\nDefining  $K:= \\ker \\varphi$ and  $\\Lambda:= (\\pi_5 \\times \\ensuremath{\\mathbb{Z}}^4)\/ (K \\times \\{0\\}) \\cong \\ensuremath{\\mathbb{Z}}^4 \\times \\ensuremath{\\mathbb{Z}}^4$, we get the exact sequence\n\\begin{equation}\\label{prop1}\n1 \\rightarrow \\Lambda \\rightarrow \\pi_1(S) \/ K \\rightarrow (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^5 \\rightarrow 1.\n\\end{equation}\n\n\\begin{prop} \n$P:=(\\pi_1(S) \/ K)^{ab} \\cong  \\ensuremath{\\mathbb{Z}}\/4\\ensuremath{\\mathbb{Z}} \\times (\\ensuremath{\\mathbb{Z}} \/2 \\ensuremath{\\mathbb{Z}})^4$.\n\\end{prop}\n\n\\begin{proof}\nWe first verify condition ($\\#\\#$) for the exact sequence (\\ref{prop1}). \n\nFor this we recall the description of  the action of $G:= \\langle g_1,g_2,g_3,g_4,g_5 \\rangle \\cong (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^5$ on \n$T:=E_1 \\times E_2\\times E_3 \\times E_4$:\n\\begin{itemize}\n\\item[] $ g_1(z_1,z_2,z_3,z_4) = \n(-z_1+\\frac 12, z_2 + \\frac 12, z_3,z_4)$,\n\\item[] $g_2(z_1,z_2,z_3,z_4) \n= (z_1, -z_2 + \\frac 12, z_3 +\\frac 12,-z_4+ \\frac 12)$,\n\\item[] $g_3(z_1,z_2,z_3,z_4) = (z_1+\\frac 12, z_2, -z_3 + \\frac 12,-z_4  + \n\\frac 12)$,\n\\item[]  $g_4(z_1,z_2,z_3,z_4) = (z_1, z_2, -z_3,-z_4)$,\n\\item[] $g_5(z_1,z_2,z_3,z_4) = (z_1+\\frac{\\tau_1}{2}, \nz_2+\\frac{\\tau_2}{2}, \nz_3+\\frac{\\tau_3}{2},z_4+\\frac{\\tau_4}{2})$.\n\\end{itemize}\n \n Observe that in this situation $\\Lambda = \\pi_1 (T) \\subset \\ensuremath{\\mathbb{C}}^4$.\n \n\nWe claim now that for each generator $\\lambda \\in  \\{\\tau_1 e_1, \\tau_2 e_2, \\tau_3 e_3, \\tau_4e_4, e_1,  e_2,e_3, e_4 \\}$ there is a $g \\in G$\n such that $g \\lambda g^{-1} = - \\lambda$.\n \n  For this it suffices to find, for each $k \\in \\{1,2,3,4\\}$, an element $g \\in G$ whose linear action has $e_k$ as $-1$ eigenvector.\n  \n But this is obvious (e.g. $(g_1g_3, g_1g_2,g_4,g_4)$ for $k=(1,2,3,4)$). \n \n Lemma \\ref{action} implies now that, setting $P':=\\pi_1(S)\/ K$, then  \n $$\n 2\\Lambda \\subset [P', P'].\n $$\nDenoting by $\\gamma_1, \\ldots , \\gamma_5$ lifts to $P'$ of the generators $g_1, \\ldots, g_5$ of $G$, we know that\n$$\nP' = \\langle \\gamma_1, \\ldots , \\gamma_5, \\lambda_1, \\ldots , \\lambda_4 \\rangle.\n$$\nMoreover, $[\\gamma_i, \\lambda_j] = \\pm 2\\lambda_j$, whence by remark \\ref{reduction}, we have an exact sequence \n$$1 \\ensuremath{\\rightarrow} \\Lambda \/2 \\Lambda \\ensuremath{\\rightarrow} P'':=P'\/2 \\Lambda \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n$$\nand $P = (P')^{ab}=(P'')^{ab} = P'' \/ \\langle [\\gamma_i, \\gamma_j], 1 \\leq i,j \\leq 5\\rangle$.\n\nA straightforward computation shows the following:\n\n$[\\gamma_1,\\gamma_2] = \\begin{pmatrix}\n0\\\\1\\\\0\\\\0\n\\end{pmatrix}$, $[\\gamma_1,\\gamma_3] = \\begin{pmatrix}\n-1\\\\0\\\\0\\\\0\n\\end{pmatrix}$, $[\\gamma_1,\\gamma_4] = 0$, $[\\gamma_1,\\gamma_5] = \\begin{pmatrix}\n- \\tau_1\\\\0\\\\0\\\\0\n\\end{pmatrix}$,\n\n$[\\gamma_2,\\gamma_3] = \\begin{pmatrix}\n0\\\\0\\\\1\\\\0\n\\end{pmatrix}$, $[\\gamma_2,\\gamma_4] = \\begin{pmatrix}\n0\\\\0\\\\1\\\\1\n\\end{pmatrix}$, $[\\gamma_2,\\gamma_5] = \\begin{pmatrix}\n0\\\\-\\tau_2\\\\0\\\\-\\tau_4\n\\end{pmatrix}$, $[\\gamma_3,\\gamma_4] = \\begin{pmatrix}\n0\\\\0\\\\1\\\\1\n\\end{pmatrix}$,\n\n$[\\gamma_3,\\gamma_5] = \\begin{pmatrix}\n0\\\\0\\\\-\\tau_3\\\\-\\tau_4\n\\end{pmatrix}$, $[\\gamma_4,\\gamma_5] = \\begin{pmatrix}\n0\\\\0\\\\-\\tau_3\\\\-\\tau_4\n\\end{pmatrix}$.\n\nThis immediately implies that \n$$\n\\tilde{\\Lambda} \/  \\langle [\\gamma_i, \\gamma_j] \\rangle \\cong \\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}}\n$$\nhence we get the exact sequence \n$$\n1\\ensuremath{\\rightarrow} \\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}} \\ensuremath{\\rightarrow} P \\ensuremath{\\rightarrow} G = (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^5 \\ensuremath{\\rightarrow} 1.\n$$\n\nSince \n$$\ng_5^2 = \\begin{pmatrix}\n\\tau_1\\\\ \\tau_2 \\\\ \\tau_3\\\\ \\tau_4\n\\end{pmatrix},\n$$\nit follows that $P \\cong \\ensuremath{\\mathbb{Z}}\/4\\ensuremath{\\mathbb{Z}} \\times (\\ensuremath{\\mathbb{Z}} \/2 \\ensuremath{\\mathbb{Z}})^4$.\n\n\\end{proof}\n\nFinally we prove\n\\begin{prop}\nThe natural surjective morphism \n$$H_1(S, \\ensuremath{\\mathbb{Z}}) = \\pi_1(S)^{ab} \\onto P\n$$ is an isomorphism.\n\\end{prop}\nThis implies theorem \\ref{homology}.\n\n\\begin{proof}\nLet $\\Gamma:= \\pi_1(S)$. Then from the exact sequence\n$$\n1 \\rightarrow H:=\\mathbb{Z}^4 \\times \\pi_5 \\rightarrow \\Gamma\n\\rightarrow G \\cong (\\mathbb{Z}\/2 \\mathbb{Z})^5\n\\rightarrow 1,\n$$\nwe get by remark \\ref{reduction} the exact sequence \n$$\n1 \\ensuremath{\\rightarrow} H^{ab} \\cong \\ensuremath{\\mathbb{Z}}^{10} \\times \\ensuremath{\\mathbb{Z}}^4 =: \\Lambda_1 \\oplus \\Lambda_2 \\ensuremath{\\rightarrow} \\Gamma' \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n$$\nand $\\Gamma^{ab} =(\\Gamma')^{ab}$. \n\nBy lemma \\ref{action} we get an exact sequence \n$$\n\\ensuremath{\\mathbb{Z}}^{10} \\times (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4 \\ensuremath{\\rightarrow} \\Gamma'' \\rightarrow G \\rightarrow 1,\n$$\nwhere again $(\\Gamma'')^{ab} = \\Gamma^{ab}$. \n\nNote that $g_1$ acts trivially on the curve $D$ of genus 5, whence it acts trivially on $H_1(D) \\cong \\ensuremath{\\mathbb{Z}}^{10}$. \n\nMoreover, we have seen that the commutators $[\\ga''_1, \n\\ga''_i]$, $2 \\leq i \\leq 5$, span a subspace $V$ of rank 3 in $\\Lambda_2 \/ 2 \\Lambda_2 \\cong (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4$. Therefore we get (after moding out by $V$) an exact sequence\n$$\n \\ensuremath{\\mathbb{Z}}^{10} \\times \\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}} \\ensuremath{\\rightarrow} \\Gamma''' \\rightarrow G \\rightarrow 1,\n$$\nwhere $(\\Gamma''')^{ab} = \\Gamma^{ab}$. After dividing by $\\ensuremath{\\mathbb{Z}}\/2\\ensuremath{\\mathbb{Z}}$, we obtain the exact sequence \n$$\nH_1(D, \\ensuremath{\\mathbb{Z}}) \\rightarrow \\pi := \\Gamma'''\/(\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})  \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1.\n$$\nWe finally get the following commutative diagram with exact rows\n\\begin{equation}\\label{diagram}\n\\xymatrix{\n\\ensuremath{\\mathbb{Z}}\/ 2 \\ensuremath{\\mathbb{Z}}\\ar[r]&\\Gamma'''\\ar[d] \\ar[r] & \\pi\\ar[d] \\ar[r]&1\\\\\n\\ensuremath{\\mathbb{Z}}\/ 2 \\ensuremath{\\mathbb{Z}}\\ar[r]&(\\Gamma''')^{ab}\\ar[r]&(\\pi)^{ab} = (\\pi_1^{orb})^{ab} \\oplus \\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}} \\cong (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^5\\ar[r]&1,\n}\n\\end{equation}\n(cf. the exact sequence (\\ref{orbifold})). \n\nThis implies that $H_1(S) = \\Gamma ^{ab} = (\\Gamma''')^{ab}$ has cardinality at most $2 \\cdot 2^5 = 2^6$. Since $|P| = 2^6$, \nthe surjective morphism $H_1(S, \\ensuremath{\\mathbb{Z}}) \\onto P$ is then an isomorphism. \n\\end{proof}\n\n\n\n\\section{Weak rigidity  of Inoue surfaces}\n\nIn this section we  shall prove the following:\n\n\\begin{theo}\\label{homotopy}\n    Let $S'$ be a smooth complex projective surface which is\nhomotopically equivalent to an Inoue surface (with $K^2 = 7$ and\n$p_g=0$). Then $S'$ is an Inoue  surface.\n\nThe same consequence holds under the weaker assumptions that $S'$ has the same fundamental group of an Inoue surface,\nand that (2) (SAME HOMOLOGY) of theorem \\ref{special diagonal} holds.\n\n\\end{theo}\n\n\\begin{rem}\n   It is clear from the definition that  $S$ is a diagonal CIT (classical Inoue   type) manifold.\n   \n   We can therefore apply theorem \\ref{special diagonal} to this special case.\n    In this special case, we are going to see that the groups $\\hat{G}_i = \\Lambda'_i \\rtimes G_{2,i}$, $i=1,2$, are obtained simply by taking $  \\Lambda'_i : = \\frac{1}{2}  \\Lambda_i$, and $ G_{2,i} : = \\{ \\pm 1\\}$.\n    \n     We shall indeed view things geometrically, as follows.\n   \n  Recall  that there is an exact sequence\n$$ 1 \\rightarrow \\mathbb{Z}^4 \\times \\pi_5 \\rightarrow \\pi_1(S)\n\\rightarrow G \\cong (\\mathbb{Z}\/2 \\mathbb{Z})^5\n\\rightarrow 1,\n$$ where $\\pi_5$ denotes the fundamental group of a compact curve\n of genus five.\n\\end{rem}\n\n\\noindent Let $\\hat{X} \\subset E_1 \\times E_2 \\times D$ be the\n\\'etale $(\\ensuremath{\\mathbb{Z}} \/2\\ensuremath{\\mathbb{Z}})^5$-covering of $S$. Observe that $H_1:=\n\\langle g_2,g_3,g_4,g_5 \\rangle$ acts trivially on $H^0(E_1,\n\\Omega^1_{E_1})$. This shows that $q(\\hat{X} \/G) \\geq 1$, and\n$$ q(\\hat{X} \/H_1) = 1 \\ \\ \\iff \\ \\ D\/H_1 \\cong \\mathbb{P}^1.\n$$ But this is obvious, since $D\/G = D\/H_1$.\n\nNow, consider instead $H_2 := \\langle g_1,g_3,g_4,g_5 \\rangle$, which\nacts trivially on $H^0(E_2, \\Omega^1_{E_2})$.\nTherefore\n$$ q(\\hat{X} \/H_2) = 1 \\ \\ \\iff \\ \\ D\/H_2 \\cong \\mathbb{P}^1.\n$$\n\n$D\/H_2 \\cong \\mathbb{P}^1$ follows from the following lemma\n\\ref{quot}, since $H_2$ contains three elements having fixed\npoints on $D$ (in fact, the elements  $(-z_3 + \\frac 12,-z_4 + \\frac\n12)$, $(-z_3 + \\frac{\\tau_3}{2},-z_4 +\n\\frac{\\tau_4}{2})$,$(-z_3 + \\frac 12 +\\frac{\\tau_3}{2},-z_4 + \\frac \n12+\\frac{\\tau_4}{2})$).\n\nTherefore we have \nseen:\n\n\\begin{prop}\\label{irreg}\n  Let $S:=\\hat{X} \/ G$ be an Inoue \nsurface. Then there are subgroups \n$H_1, H_2 \\leq G$ of index 2, such \nthat $q(\\hat{X}\/H_1) =\nq(\\hat{X}\/H_2) = 1$.\n\nLet $S'$ be a smooth \ncomplex projective surface which has the same \nfundamental group as \n$S$. Then, denoting by $\\hat{X}'$ the\ncorresponding \\'etale \n$G$-covering of $S'$ we have:\n\\begin{itemize}\n  \\item there is a \nsmooth curve $D'$ of genus 5 and a surjective \nhomomorphism $\\hat{X}' \n\\rightarrow D'$;\n\\item there are two index two subgroups $H_1, H_2$ \nof $G$ such that \n$q(\\hat{X}\/H_1) = q(\\hat{X}\/H_2) = \n1$.\n\\end{itemize}\n\n\\end{prop}\n\n\n\\begin{lemma}\\label{quot}\n\nLet $D \\rightarrow \\mathbb{P}^1$ be the maximal $\\Gamma \\cong\n(\\mathbb{Z} \/ 2 \\mathbb{Z})^4$ -covering branched in five\npoints. Then:\n\n1) there are exactly 5 subgroups $H \\leq \\Gamma$, $H \\cong\n(\\mathbb{Z}\/2 \\mathbb{Z})^3$ containing exactly one element\nhaving fixed points on $D$ ($\\implies$ $g(D\/H) = 1$);\n\n2) there are exactly 10 subgroups $H \\leq \\Gamma$, $H \\cong\n(\\mathbb{Z}\/2 \\mathbb{Z})^3$ containing exactly three elements\nhaving fixed points on $D$ ($\\implies$ $g(D\/H) = 0$).\n\\end{lemma}\n\n\\begin{proof} Let $D \\rightarrow \\mathbb{P}^1$ be the maximal $\\Gamma\n\\cong (\\mathbb{Z} \/ 2 \\mathbb{Z})^4$ -covering\nbranched in five points $p_1, \\ldots p_5$, which determines a\nsurjective homomorphism\n$$\n\\varphi \\colon \\pi_1(\\mathbb{P}^1 \\setminus \\{p_1, \\ldots p_5 \\})\n\\rightarrow \\Gamma.\n$$\n\nWe denote $\\varphi(\\gamma_i)$, where $\\gamma_i$ is a geometric loop\naround $p_i$ by $e_i$. Then $e_1, e_2,e_3, e_4$ is a\n$\\mathbb{Z}\/ 2\\mathbb{Z}$ - basis of $\\Gamma$ and $e_1 + e_2 +e_3+e_4 = e_5$.\n\n\\begin{claim}\n   Let $H \\cong (\\ensuremath{\\mathbb{Z}} \/2 \\ensuremath{\\mathbb{Z}})^3 \\leq \\Gamma$. Then either there is a\nunique $i \\in \\{ 1, \\ldots , 5 \\}$ such that $e_i \\in H$ (and\n$e_j \\notin H$ for $j \\neq i$, or there is a subset $\\{i,j,k \\}\n\\subset \\{ 1, \\ldots , 5 \\}$ such that $e_i, e_j, e_k \\in H$ (and the\nother two $e_l$'s are not in $H$).\n\\end{claim}\n\n{\\em Proof of the claim.} It is clear that $H$ contains at\nleast one of the $e_i$'s, otherwise $D \\rightarrow D\/H$ is\n\\'etale. By Hurwitz' formula, we get\n$$ 8 = 2g(D) -2 = |H|(2(g(D\/H)-2) = 8(2g(D\/H)-2),\n$$ a contradiction. Since any four of the $e_i$'s are linearly\nindependent, $H$ can contain at most three of them.\n\nAssume now that there are $i \\neq j$, such that $e_i, e_j \\in H$.\nW.l.o.g. we can assume $e_1, e_2 \\in H$. Then $H = \\langle e_1,\ne_2, \n\\lambda_1e_1 +\\lambda_2e_2 +\\lambda_3e_3 +\\lambda_4e_4 \\rangle$, \n\n$\\lambda_i \\in \\{0,1\\}$. Note that $e_1, e_2,\n\\lambda_1e_1 \n+\\lambda_2e_2 +\\lambda_3e_3 +\\lambda_4e_4$ are linearly \nindependent \nif and only if $(\\lambda_3,\n\\lambda_4) \\neq (0,0)$. \nMoreover,\n\\begin{itemize}\n\\item $e_3 \\notin H  \\iff \n(\\lambda_3,\\lambda_4) \\neq (1,0)$,\n\\item $e_4 \\notin H  \\iff \n(\\lambda_3,\\lambda_4) \\neq (0,1)$,\n\\item $e_5 \\notin H  \\iff \n(\\lambda_3,\\lambda_4) \\neq (1,1)$.\n\\end{itemize} This shows: if $H$ \ncontains two of the $e_i$'s then \nalso a third one.\n\n\\QED\n\nNow, \nif $H$ contains three of the $e_i$'s, say $e_i, e_j,e_k$, then \n$H = \n\\langle e_i, e_j, e_k \\rangle$, and there are exactly\n$\\binom{5}{3} = \n10$ such subgroups.\n\nThe remaining $5$ subgroups $H \\cong (\\ensuremath{\\mathbb{Z}} \/ 2 \n\\ensuremath{\\mathbb{Z}})^3$ of $\\Gamma$ \ncontain therefore exactly one of the \n$e_i$'s.\n\nAssume now that\n$$ H_1 := \\langle e_1, e_2, e_3 \\rangle, \\ \n\\ H_2:= \\langle e_1+e_2, \ne_1+e_3, e_1+e_4 \\rangle.\n$$\n\nThen it \nremains to show that $g(D\/H_1)=0$ and $g(D\/H_2) = 1$.\n\nObserve that \n$H_2 = \\langle e_5 \\rangle \\oplus \\langle e_1 +e_2, \ne_1+e_3 \n\\rangle$, and $H':=\\langle e_1 +e_2, e_1+e_3\n\\rangle$ acts freely on \n$D$. Therefore $g(D\/H') = 2$ and $e_5$ acts \non $D\/H'$ having two \nfixed points. By Hurwitz' formula this\nimplies that $g(D\/H) \n=1$.\n\nNow, $g(D\/ \\langle e_1 \\rangle) = 1$ and $H_1\/ \\langle e_1 \n\\rangle \n\\cong \\langle e_2, e_3\\rangle$ acts with fixed points on\n$D\/ \n\\langle e_1 \\rangle$. This shows that $D\/H_1 \\cong \n\\ensuremath{\\mathbb{P}}^1$.\n\n\\end{proof}\n\nNow we are ready to finish the proof of  \\ref{homotopy}.\n\n\\begin{proof}(of thm. \\ref{homotopy})\n\nLet $S'$ be a \nsmooth complex projective surface which is homotopically equivalent \nto an Inoue surface $S$ with $K_S^2 =7$ and\n$p_g=0$. \n\nIn particular \n$\\pi_1 (S) \\cong \\pi_1(S')$ and we take the \\'etale $G:=(\\ensuremath{\\mathbb{Z}} \/ \n2\\ensuremath{\\mathbb{Z}})^5$- covering $\\hat{X}'$, which\nis homotopically equivalent to \n$\\hat{X}$ (the corresponding covering \nof the Inoue surface $S$).  By \nproposition \\ref{irreg} we\nknow that $\\hat{X}'$ admits a morphism to \na curve $D'$ of genus 5, \nand there are subgroups $H_1, H_2 \\leq G$ \nof index 2 such\nthat $X_i:=\\hat{X}'\/H_i$ has irregularity \none.\n\nTherefore there are elliptic curves\n$E_1', E_2'$ and \nmorphisms\n$$\n\\hat{X}' \\rightarrow X_i \\rightarrow E'_i.\n$$ By the \nuniversal property of the Albanese map we get a commutative \ndiagram\n\n\\begin{equation}\\label{diagr}\n\\xymatrix{\n\\hat{X}'\\ar[r]\\ar[d]&E_1' \n\\times E_2' \\times \\Jac(D')\\\\\n\\Alb(\\hat{X}')\\ar_{\\psi}[ru]&\\\\ \n}\n\\end{equation}\n\n\\begin{lemma} Let $E_i' = \\mathbb{C}\/\\Lambda_i'$ \nand denote by \n$\\Lambda_i:= 2\\Lambda_i'$. Then $\\psi$ corresponds to \n$$\nH_1(D', \\ensuremath{\\mathbb{Z}}) \\times \\Lambda_1 \\times \\Lambda_2 \\subset H_1(D', \n\\ensuremath{\\mathbb{Z}}) \n\\times \\Lambda_1' \\times \\Lambda_2'.\n$$\n\nIn particular $E_i:= \n\\mathbb{C}\/\\Lambda_i = E_i'$, $\\psi$ \nrestricted to $E_i$ is \nmultiplication by 2, and\n$\\Alb(\\hat{X}') = \\Jac(D') \n\\times E_1 \\times E_2.$\n\n\\end{lemma} \n\n By theorem \\ref{special diagonal} we have that\n$\\varphi \\colon \\hat{X}' \\rightarrow \n E_1 \\times \nE_2 \\times D' $ is a birational morphism onto its image \n$W'\n\\subset  E_1 \\times \nE_2 \\times D'  : = C_1 \\times C_2 \\times C_3$.\n\n In fact, since the fundamental group of $X'$ is isomorphic to the\none of a Inoue surface, it follows by lemma \\ref{affine} that $G$ acts on $Z'$ as for an Inoue surface, hence\n there is no effective divisor $\\De$ of \nnumerical type $(1,1,2)$ which is invariant by the action of the group $G$, as it is easy to verify.\n\nTherefore  $W'$ has homology class $2F_1 + 2F_2 \n+ 4F_3$, where $F_i$ is the fibre over a point in the i-th curve, $C_i$\nand $W'$ has rational double points as singularities.\n\n\nThe linear equivalence class of $W'$ is invariant for the group action. It is the sum of three classes $\\xi_i$ which\nare respective pull backs from the projection\nonto the i-th curve $C_i$. Hence each  class $\\xi_i$ is invariant for the action of $G$ on $C_i$,\nhence $\\xi_i$ is the pull back from the quotient of $C_i$ by the group $G_i$, projection of $G$ into the automorphism group\nof $C_i$.\n\nBy our lemmas, \\ref{genus1} and \\ref{genus5}, these quotients are rational curves, hence\nwe conclude that the linear equivalence class of the divisor $W'$ is the same as the one for\nan Inoue surface $S$.\n\nIt remains to show that $W'$ is given by Inoue's equations, i.e., if we consider the genus 5 curve $D'$ as a hypersurface \n$$\nD' := \\{ z_3,z_4)) \\in E_3 \\times E_4 \\ |  \\ \\sL_3(z)\\sL_4(z_4) =\nb_3b_4\\}, \n$$\nthen \n\\begin{multline}\nW' := \\{ (z_1,z_2,z_3,z_4)) \\in E_1 \\times  \\ldots \\times E_4 \\ | \\ \\sL_1(z_1)\n\\sL_2(z_2)\\sL_3(z_3)  = b_1b_2b_3, \\\\\n \\sL_3(z_3)\\sL_4(z_4) =\nb_3b_4\\}.\n\\end{multline}\nFor this consider the subgroup $H:=\\langle g_1,g_2,g_3,g_4 \\rangle \\leq G$. Then there is a $H$-invariant divisor with class $2F_1+2F_2 +4F_3$ in $Z':=E_1 \\times E_2 \\times D'$. Therefore $H^0(Z', \\ensuremath{\\mathcal{O}}_Z'(2F_1+2F_2+4F_3)) \\cong \\ensuremath{\\mathbb{C}}^8$ is an $H$-module which decomposes by K\\\"unmeth's formula and the results in section \\ref{section2} as follows:\n\n\\begin{multline}\nH^0(Z', \\ensuremath{\\mathcal{O}}_Z'(2F_1+2F_2+4F_3)) \\cong \\\\\n\\cong H^0(E_1 \\times E_2 \\times D', p_1^*(\\ensuremath{\\mathcal{O}}_{E_1}(2[0]))\n\\otimes p_2^*(\\ensuremath{\\mathcal{O}}_{E_2}(2[0]))\\otimes p_3^*(\\ensuremath{\\mathcal{O}}_{D'}(2,0))) \\cong \\\\\n\\cong V_1 \\otimes V_2 \\otimes V_3 \\cong (V_1^{++++} \\oplus V_1^{-+-+}) \\otimes  (V_2^{++++} \\oplus V_2^{--++}) \\otimes (V_3^{++++} \\oplus V_3^{+--+}) \\cong \\\\\n\\cong V^{++++} \\oplus V^{+--+} \\oplus V^{--++} \\oplus V^{-+-+},\n\\end{multline}\nwhere each of the four summands in the last line is isomorphic to $\\ensuremath{\\mathbb{C}}^2$. In fact, we have\n\n$$V^{++++} = (V_1^{++++} \\otimes V_2^{++++} \\otimes V_3^{++++} ) \\oplus (V_1^{-+-+} \\otimes V_2^{--++} \\otimes V_3^{+--+} ),\n$$\n$$V^{+--+} = (V_1^{++++} \\otimes V_2^{++++} \\otimes V_3^{+--+} ) \\oplus (V_1^{-+-+} \\otimes V_2^{--++} \\otimes V_3^{++++} ),\n$$\n$$V^{--++} = (V_1^{++++} \\otimes V_2^{--++} \\otimes V_3^{++++} ) \\oplus (V_1^{-+-+} \\otimes V_2^{++++} \\otimes V_3^{+--+} ),\n$$\n$$V^{-+-+} = (V_1^{-+-+} \\otimes V_2^{++++} \\otimes V_3^{++++} ) \\oplus (V_1^{++++} \\otimes V_2^{--++} \\otimes V_3^{+--+} ).\n$$\nThe equations of the hypersurfaces in the above pencils are then:\n$$\nW'_1(c) := \\{ c = \\sL_1(z_1) \\sL_2(z_2) \\sL_3(z_3) \\},\n$$\n$$\nW'_2(c) := \\{ c = \\sL_1(z_1) \\sL_2(z_2) \\frac{1}{\\sL_3(z_3)} \\},\n$$\n$$\nW'_3(c) := \\{ c = \\sL_1(z_1) \\frac{1}{\\sL_2(z_2)} \\sL_3(z_3) \\},\n$$\n$$\nW'_4(c) := \\{ c = \\frac{1}{\\sL_1(z_1)} \\sL_2(z_2) \\sL_3(z_3) \\}.\n$$\n\nThis shows that after possibly replacing one of the elliptic curves $E_i$ with parameter $a_i$ by the elliptic curve $E'_i$ with parameter $\\frac{1}{a}$ we can w.l.o.g. assume that the pencil of $H$-invariant hypersurfaces in $Z'$ is given by the equation $\\{ c = \\sL_1(z_1) \\sL_2(z_2) \\sL_3(z_3) \\}$.\n\nNow, we consider $g_5$. It is easy to see that $g_5(W'_1(c)) \\equiv W'_1(c)$, and if $W'_1(c) = div(s)$ for $s \\in V^{++++}$, then also $g_5(W'_1(c)) = div(s')$ for $s' \\in V^{++++}$. Therefore $g_5$ is an involution on $\\ensuremath{\\mathbb{P}}^1 := \\ensuremath{\\mathbb{P}}(V^{++++})$, which is obviously non trivial,  whence $g_5$ has exactly two fixed points. Therefore there are exactly two $G$-invariant divisors in the pencil $W'_1(c)$. On the other hand,\n$W'_1(b_1b_2b_3)$ and $W'_1(-b_1b_2b_3)$ are $G$-invariant. \n\nThis shows that $W'$ is of the desired  form, hence $X'$ is the canonical model of an Inoue surface.\n\n\\end{proof}\n\n\n\\section{Inoue surfaces as bidouble \ncovers and $H^1(S,\\Theta_S)$}\n\nThe aim of this section is to show the \nfollowing \n\\begin{theo}\nLet $S$ be an Inoue surface with $K_S^2 = 7$. \nThen:\n$$\nh^1(S, \\Theta_S) = 4, \\ \\  h^2(S, \\Theta_S) = \n8.\n$$\n\\end{theo}\n\nTo prove this result we resort to a result of \n\\cite{mlp}, where   Inoue surfaces are constructed as bidouble covers of the \nfour nodal cubic.\n\nWe briefly recall their description here, for \ndetails we refer to \n\\cite{mlp}, example 4.1 (we keep their notation, even if slightly inconvenient).\n\nWe consider a complete \nquadrilateral $\\Lambda$ in $\\ensuremath{\\mathbb{P}}^2$  and \ndenote the vertices by $P_1, \\ldots,\nP_6$.\n\nWe have labeled the \nvertices in a way that\n\\begin{itemize}\n  \\item the intersection point \nof the line $\\overline{P_1P_2}$ and the line \n$\\overline{P_3P_4}$ is $P_5$,\n\\item the \nintersection point of $\\overline{P_1P_4}$ and $\\overline{P_2P_3}$ is \n$P_6$.\n\\end{itemize}\n\nLet $Y \\rightarrow \\ensuremath{\\mathbb{P}}^2$ be the blow up in \n$P_1, \\ldots , P_6$, \ndenote by $L$  the total transform of a line in \n$\\ensuremath{\\mathbb{P}}^2$, let $E_i$, $1\n\\leq i \\leq6$, be the exceptional curve lying over \n$P_i$.\nMoreover, we denote by $S_i$, $1 \\leq i \\leq 4$, the strict \ntransforms on $Y$ of the sides $S_i : = \\overline{P_i P_{i+1}}$ for $ 1\\leq i \\leq 3$,\n$S_4 : = \\overline{P_4 P_{1}}$, of the quadrilateral $\\Lambda$.\n\nThe geometry of the situation is that the four (-2) curves $S_i$ come from the resolution\nof the 4 nodes of the cubic surface $\\Sigma$ which is the anticanonical image of $Y$,\nand the curves $E_i$ are the strict transforms of the 6 lines in $\\Sigma$ connecting pairs of nodal points.\n\nThe surface $\\Sigma$ contains also a triangle of lines (joining the midpoints of opposite edges\nof the tetrahedron with sides the lines corresponding to the curves $E_i$. These are the 3 strict transforms \n $\\Delta_1$, $\\Delta_2$,\n$\\Delta_3$ of the three diagonals of the complete quadrilateral $\\Lambda$. $\\Delta_1$\nis the strict transform of $\\overline{P_1P_3}$, $\\Delta_2$ of $\\overline{P_2P_4}$ and $\\Delta_3$\nof $\\overline{P_5P_6}$.\n\nFor each line $\\De_i$ in the cubic surface  $\\Sigma$ we consider the pencil of planes containing\nthem, and the base point free pencil of residual conics, which we denote by $| f_i|$. Hence we have\n$$| f_i| = |(-K_Y) -  \\De_i| , \\ \\  \\De_i + f_i \\equiv (-K_Y) .$$\n\nIn the plane realization we have:\n\\begin{itemize}\n  \\item $f_1$ is the strict \ntransform on $Y$ of a general element of\nthe pencil of conics $\\Gamma_1$ through\n$P_2,P_4,P_5,P_6$,\n\\item $f_2$ is the strict transform on $Y$ of a general element of\nthe pencil of conics $\\Gamma_2$ through $P_1,P_3,P_5,P_6$,\n\\item $f_3$ is the strict transform on $Y$ of a general element of\nthe pencil of conics $\\Gamma_3$ through $P_1,P_2,P_3,P_4$.\n\\end{itemize}\n\nIt is then easy to see that each curve $S_h$ is disjoint from the other curves $S_j$ ($j\\neq h$), $\\De_i$, and $f_i$\nif $f_i$ is smooth. Moreover, \n$$\\De_i \\cdot f_i = 2,  \\  \\  \\De_i \\cdot f_j = 0, i \\neq j, \\ \\ f_i ^2 = 0 , \\ \\  f_j f_i = 2,  i \\neq j . $$\n\n\n\n\\begin{definition}\\label{Inouedivisors}\n   We define the {\\em Inoue divisors} on $Y$ as follows:\n\\begin{itemize}\n\\item $D_1:= \\Delta_1 + f_2 + S_1 + S_2$, where $f_2 \\in |f_2|$ smooth;\n\\item $D_2:= \\Delta_2 + f_3$, where $f_3 \\in |f_3|$ smooth;\n\\item$D_3:= \\Delta_3 + f_1 +f_1'+ S_3 + S_4$, where $f_1, f_1' \\in\n|f_1|$ smooth.\n\\end{itemize}\n\n\\end{definition} \n\nLet $\\pi \\colon \\tilde{S} \\rightarrow Y$ be the\nbidouble covering with branch divisors $D_1, D_2, D_3$ (associated to the \n3 nontrivial elements of the Galois group.\n\n By the previous remarks we see\nthat over each $S_i$ there are two disjoint $(-1)$-curves.\nContracting these eight exceptional curves we obtain a minimal surface\nwith $p_g=0$ and $K_S^2 = 7$. By \\cite{mlp} these are exactly the Inoue surfaces.\n\n\\begin{rem}\\label{ratvar}\nWe immediately see that there is an open dense subset  in the product \n$$\n|f_1| \\times |f'_1| \\times |f_2| \\times |f_3| \\cong (\\ensuremath{\\mathbb{P}}^1)^4\n$$\nparametrizing the family of Inoue surfaces.\n\\end{rem}\n\n\n\\begin{rem}\n   The non trivial character sheaves of this bidouble cover are\n\\begin{itemize}\n   \\item $\\mathcal{L}_1 = \\ensuremath{\\mathcal{O}}_Y( - K_Y + f_1 - E_4)$;\n\\item $\\mathcal{L}_2 = \\ensuremath{\\mathcal{O}}_Y( - 2 K_Y  - E_5 - E_6)$;\n\\item $\\mathcal{L}_3 = \\ensuremath{\\mathcal{O}}_Y( - K_Y + L - E_1 - E_2  - E_3)$.\n\\end{itemize}\n\n\\end{rem}\n\n\n\\begin{lemma}\\label{inv} \\\n\\begin{itemize}\n    \\item $\\dim H^1(\\tilde{S}, \\Theta_{\\tilde{S}})^{inv} = \\dim H^1(S,\n\\Theta_S)^{inv} = 4$, \n\\item  $\\dim H^2(\\tilde{S}, \\Theta_{\\tilde{S}})^{inv} = \\dim H^2(S,\n\\Theta_S)^{inv} = 0$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nIt is well known (cf. e.g.  \\cite{cime2}) that  $H^2(\\tilde{S}, \\Theta_{\\tilde{S}})$ of the $(\\ensuremath{\\mathbb{Z}}\/2\\ensuremath{\\mathbb{Z}})^2$-covering  $\\pi \\colon \\tilde{S} \\ensuremath{\\rightarrow} Y$ decomposes as a direct sum of character spaces\n$$\nH^2(\\tilde{S}, \\Theta_{\\tilde{S}}) \\cong H^2(\\tilde{S}, \\Theta_{\\tilde{S}})^{inv} \\oplus \\bigoplus_{i=1}^3 H^2(\\tilde{S}, \\Theta_{\\tilde{S}})^i,\n$$\nand that the dimensions of the direct summands can be computed as the dimensions of global sections of logarithmic differential forms on the basis $Y$. In fact, we have:\n\\begin{multline}\\label{chars}\nh^0(Y, \\Omega^1_Y(\\log D_1, \\log D_2, \\log D_3)(K_Y)) = h^2(\\tilde{S},\\Theta_{\\tilde{S}})^{inv} = h^2(S,\\Theta_S)^{inv}; \\\\\nh^0(Y, \\Omega^1_Y(\\log D_i)(K_Y + \\mathcal{L}_i))) = h^2(\\tilde{S},\\Theta_{\\tilde{S}})^i = h^2(S,\\Theta_S)^i, \\forall i \\in \\{1,2,3\\}.\n\\end{multline}\nNote that $|-K_{Y}| = |3L - \\sum_{i=1}^6 E_i|$ \nis non empty \nand does not have a fixed part. Therefore there is a \nmorphism\n$\\ensuremath{\\mathcal{O}}_{Y}(K_{Y}) \\ensuremath{\\rightarrow} \\ensuremath{\\mathcal{O}}_{Y}$, \nwhich is not \nidentically zero on any component of the \n$D_i$'s.\n\n\n\nWe get the \ncommutative diagram with exact \nrows\n\\begin{equation}\\label{diagram2}\n\\xymatrix{ 0\\ar[r]& \n\\Omega^1_{Y}(K_{Y}) \\ar[d] \n\\ar[r]&\\Omega^1_{Y}((\\log D_1, \\log D_2, \\log D_3)(K_{Y})\\ar[d]\\ar[r]\n&\\oplus_{i=1}^3 \n\\ensuremath{\\mathcal{O}}_{D_i}(K_{Y}) \\ar[d] \\ar [r]&0\\\\ \n0\\ar[r]&\\Omega^1_{Y} \\ar[r]&\\Omega^1_{Y}((\\log D_1, \\log D_2, \\log D_3) \n\\ar[r] &\\oplus_{i=1}^3 \\ensuremath{\\mathcal{O}}_{D_i}  \\ar [r]&0.}\n\\end{equation}\n\n From \nthis we get the commutative \ndiagram with injective vertical arrows\n\\begin{equation*}\n\\xymatrix{\n\\ensuremath{\\mathbb{C}}^2 \\oplus 0 \\oplus\\ensuremath{\\mathbb{C}}^2 \n\\cong&H^0(Y,\\oplus_{i=1}^3\n\\ensuremath{\\mathcal{O}}_{D_i}(K_{Y}))\\ar[r]^{\\delta}\\ar[d]^{\\psi_2}\\ar[dr]^{\\varphi}& \nH^1(Y,\n\\Omega^1_{Y}(K_{Y})) \\ar[d] \\\\\n\\ensuremath{\\mathbb{C}}^4 \n\\oplus \\ensuremath{\\mathbb{C}}^2 \\oplus\\ensuremath{\\mathbb{C}}^5 \\cong &H^0(Y,\\oplus_{i=1}^3 \n\\ensuremath{\\mathcal{O}}_{D_i})\\ar[r]^{\\psi_1} &H^1(Y,\n\\Omega^1_{Y}) \n.}\n\\end{equation*} \n\nA usual argument shows that (see \\cite{catjdg}) $\\delta$ is injective. In fact, the Chern classes of $S_1,S_2,S_3,S_4$ are linearly independent, hence  $\\varphi$ is injective, which implies that also\n$\\delta$ is \ninjective. Therefore\n$$ h^0(\\Omega^1_{Y}((\\log D_1, \\log D_2, \\log D_3)(K_{Y})) \n= \nh^2(\\Theta_{\\tilde{S}})^{inv} = h^2(\\Theta_S)^{inv} = \n0.\n$$\n\nTherefore\n\\begin{multline}\n  h^1(\\Theta_{\\tilde{S}})^{inv} = - \n\\chi(\\Omega^1_{Y}((\\log D_1, \\log D_2, \\log D_3)(K_{Y})) = \\\\\n=\n-(\\chi(\\Omega^1_{Y}(K_{Y}) + \\chi(\\oplus_{i=1}^3 \n\\ensuremath{\\mathcal{O}}_{D_i}(K_{Y}))).\n\\end{multline}\n\nAn easy calculation shows now that \n$\\chi(\\oplus_{i=1}^3 \n\\ensuremath{\\mathcal{O}}_{D_i}(K_{Y}))) = 0$, \nwhereas\n$\\chi(\\Omega^1_{Y}(K_{Y})) = - 4$.\n\\end{proof}\n\nWe prove now\n\n\\begin{prop}\\label{h1theta}  \\\n\\begin{enumerate}\n\\item $h^0(Y, \\Omega^1_Y(\\log D_1)(K_Y + \\mathcal{L}_1)) = \nh^2(\\tilde{S}, \\Theta_{\\tilde{S}})^1 = h^2(S, \\Theta_S)^1 \\leq 2$;\n\\item $h^0(Y, \\Omega^1_Y(\\log D_2)(K_Y + \\mathcal{L}_2)) = \nh^2(\\tilde{S}, \\Theta_{\\tilde{S}})^2 = h^2(S, \\Theta_S)^2 \\leq 3$;\n\\item $h^0(Y, \\Omega^1_Y(\\log D_3)(K_Y + \\mathcal{L}_3)) = \nh^2(\\tilde{S}, \\Theta_{\\tilde{S}})^3 = h^2(S, \\Theta_S)^3 \\leq 3$.\n\\end{enumerate}\nIn particular, we get $h^2(S,\\Theta_S) \\leq 8$.\n\\end{prop}\n\nFrom this result, we easily obtain the following:\n\\begin{cor}\\label{cortheta}\nLet $S$ be an Inoue surface with $K_S^2 =7$, $p_g=0$. Then:\n$$\nh^2(S, \\Theta_S) = 8, \\ \\ h^1(S, \\Theta_S)= h^1(S, \\Theta_S)^{inv} = 4.\n$$\n\n\\end{cor}\n\n\\begin{proof}[Proof of the cor..]\nWe have by proposition \\ref{h1theta} \n$$\n8 - h^1(S,\\Theta_S) \\geq h^2(S, \\Theta_S) - h^1(S,\\Theta_S) = \\chi(\\Theta_S) = 2K_S^2 - 10 \\chi(S) = 4,\n$$\nwhence $h^1(S, \\Theta_S) \\leq 4$. On the other hand by lemma \\ref{inv} we know that\n$$\n4 = h^1(S, \\Theta_S)^{inv} \\leq h^1(S, \\Theta_S)\n$$\nand the assertions of the corollary follow.\n\\end{proof}\n\\begin{proof}[Proof of proposition \\ref{h1theta}.]\n1) Recall that by definition \\ref{Inouedivisors} $D_1$ is the \ndisjoint union of the four curves $\\Delta_1, f_2, S_1, S_2$, and\n$$\nK_Y + \\mathcal{L}_1 \\equiv  f_1 -E_4.\n$$\n\nWe consider the exact sequence (cf. e.g.  \\cite{esnaultviehweg}, p. 13)\n\\begin{multline}\\label{exseq}\n0 \\ensuremath{\\rightarrow} \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1)(-E_4) \\ensuremath{\\rightarrow} \\\\\n\\ensuremath{\\rightarrow} \\Omega^1_Y((\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2)(f_1-E_4)) \\ensuremath{\\rightarrow} \\\\\n\\ensuremath{\\rightarrow} \\Omega^1_{f_1} (\\Delta_1 +f_2 + S_1+S_2+f_1-E_4) \\ensuremath{\\rightarrow} 0.\n\\end{multline}\n\nSince\n$$\n(\\Delta_1 +f_2 + S_1+S_2+f_1-E_4)f_1 = 2+2-1,\n$$\nwe have $ \\Omega^1_{f_1} (\\Delta_1 +f_2 + S_1+S_2+f_1-E_4) \\cong \\ensuremath{\\mathcal{O}}_{\\ensuremath{\\mathbb{P}}^1}(1)$, whence \n\\begin{multline}\nh^0(Y, \\Omega^1_Y(\\log D_1)(K_Y+ \\mathcal{L}_1)) = h^0(Y, \n\\Omega^1_Y((\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2)(f_1-E_4)) \\leq \\\\\n \\leq h^0(Y, \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1)(-E_4)) + 2.\n\\end{multline}\nConsider the long exact cohomology sequence of the short exact sequence\n\n\\begin{multline}\n0 \\ensuremath{\\rightarrow} \\Omega^1_Y \\ensuremath{\\rightarrow}  \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1) \\ensuremath{\\rightarrow} \\\\\n\\ensuremath{\\rightarrow}  \\ensuremath{\\mathcal{O}}_{f_1} \\oplus \\ensuremath{\\mathcal{O}}_{f_2} \\oplus \\ensuremath{\\mathcal{O}}_{S_1} \\oplus \\ensuremath{\\mathcal{O}}_{S_2} \n\\oplus \\ensuremath{\\mathcal{O}}_{\\Delta_1} \\ensuremath{\\rightarrow} 0\n\\end{multline}\n\nSince $H^0(Y, \\Omega^1_Y)=0$, \n$H^0(Y, \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1))$ is the kernel of the connecting \nhomomorphism\n$$\n\\delta \\colon H^0(Y, \\ensuremath{\\mathcal{O}}_{f_1} \\oplus \\ensuremath{\\mathcal{O}}_{f_2} \\oplus \\ensuremath{\\mathcal{O}}_{S_1} \\oplus \\ensuremath{\\mathcal{O}}_{S_2} \n\\oplus \\ensuremath{\\mathcal{O}}_{\\Delta_1}) \\ensuremath{\\rightarrow} H^1(Y, \\Omega^1_Y).\n$$\nBy \\cite{catjdg} the image of $\\delta$ is generated by the Chern \nclasses of $ \\Delta_1, f_2,  S_1, S_2, f_1$. \n\n\n\\begin{claim}\n$\\dim \\im (\\delta) = 5$.\n\\end{claim}\n\\begin{proof}[Proof of the claim.] \nAssume that\n\\begin{equation}\\label{zero1}\n\\lambda_1 f_1 + \\lambda_2 f_2 + a_1 S_1 + a_2 S_2 + \\mu \\Delta_1\\equiv 0,\n\\end{equation}\nwhere $\\lambda_1, \\lambda_2, a_1, a_2, \\mu \\in \\ensuremath{\\mathbb{C}}$.\nIntersection with $E_4$ gives $\\lambda_1 =0$, whereas intersection with $S_i$, $i=1,2$ yells $-2a_i=0$. The equation \\ref{zero1} has become\n$$\n\\mu \\Delta_1 + \\lambda_2 f_2 =0.\n$$\n\nIntersection with e.g. $E_5$ gives $\\lambda_2 =0$, whence also $\\mu = 0$.\n\\end{proof}\n\nIt follows now that\n$$\nh^0(Y, \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1)(-E_4)) \\leq \n$$\n$$\n\\leq h^0(Y, \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1)) = 0.\n$$\n\nTherefore we have proven\n$$\nh^0(Y, \\Omega^1_Y(\\log D_1)(K_Y+ \\mathcal{L}_1)) \\leq 2.\n$$\n2) By definition \\ref{Inouedivisors} $D_2$ is the disjoint union of \nthe two curves $\\Delta_2, f_3$, and\n$$\nK_Y + \\mathcal{L}_2 \\equiv  -K_Y -E_5-E_6.\n$$\nSince\n$$\n(K_Y + 2 \\Delta_2 +(- K_Y -E_5-E_6)) \\Delta_2 = (2 \\Delta_2  -E_5-E_6) \\Delta_2 -2 < 0,\n$$\nby  lemma 5.1 of  \\cite{burniat3} we have\n$$\nH^0(Y, \\Omega^1_Y(\\log D_2)(K_Y+ \\mathcal{L}_2)) \\cong H^0(Y, \n\\Omega^1_Y(\\log f_3)(K_Y + \\mathcal{L}_2 + \\Delta_2)).\n$$\nNote that\n$$\nK_Y + \\mathcal{L}_2 + \\Delta_2 \\equiv -K_Y -E_5 - E_6 + \\Delta_2 \n\\equiv S_1 + S_2 +S_4 + S_3 + E_1 + E_3.\n$$\n\nAgain by lemma 5.1 in \\cite{burniat3} we see that\n\\begin{multline}\nH^0(Y, \\Omega^1_Y(\\log f_3)(K_Y + \\mathcal{L}_2 + \\Delta_2)) \\cong \\\\\n\\cong H^0(Y, \\Omega^1_Y(\\log f_3, \\log S_1, \\log S_2, \\log S_3, \\log \nS_4)(E_1 + E_3)).\n\\end{multline}\nSince $E_1(f_3 + S_1 +S_2 + S_3 +S_4 + E_1 +E_3) = E_3(f_3 + S_1 +S_2 \n+ S_3 +S_4 + E_1 +E_3) = 2$, we see by the same arguments (using the analogous exact sequence \\ref{exseq}) as in case \n1) that\n\n\\begin{multline}\nh^0(Y, \\Omega^1_Y(\\log f_3, \\log S_1, \\log S_2, \\log S_3, \\log \nS_4)(E_1 + E_3)) \\leq \\\\\n\\leq h^0(Y, \\Omega^1_Y(\\log f_3, \\log S_1, \\log S_2, \\log S_3, \\log \nS_4, \\log E_1, \\log E_3)) +2.\n\\end{multline}\n\\begin{claim}\n$\\dim \\langle f_3, S_1, \\ldots , S_4, E_1, E_3 \\rangle =6$.\n\\end{claim}\n\\begin{proof}[Proof of the claim.]\nNote that \n$$\nS_1+S_4-S_2-S_3 \\equiv -2E_1 + 2E_3.\n$$\nTherefore we have proven the claim, if we show that the Chern classes of $f_3, S_1, \\ldots , S_4, E_1$ are linearly independent.\n\nAssume that \n\\begin{equation}\\label{zero2}\n\\lambda f_3 + a_1 S_1 + a_2 S_2 + a_3 S_3 +a_4 S_4 + \\mu E_1\\equiv 0,\n\\end{equation}\nwhere $\\lambda, a_1, a_2, a_3,a_4, \\mu \\in \\ensuremath{\\mathbb{C}}$.\n\nIntersection with $S_2, S_3$ gives $a_2 = a_3 =0$, whereas intersection with $\\Delta_1$ yields $\\mu = 0$. We are left with the equation $\\lambda f_3 + a_1 S_1 +a_4 S_4 \\equiv 0$. Intersection with $E_5$ resp. $E_6$ implies that $a_1=0$ resp. $a_4=0$, and we conclude that also $\\lambda=0$.\n\n\\end{proof}\nTherefore we get\n$$\nh^0(Y, \\Omega^1_Y(\\log f_3, \\log S_1, \\log S_2, \\log S_3, \\log S_4, \n\\log E_1, \\log E_3)) = 1,\n$$\nand we have shown 2).\n\n3)  $D_3$ is the disjoint union of the five curves $\\Delta_3, f_1, \nf'_1,S_3, S_4$, and\n$$\nK_Y + \\mathcal{L}_3 \\equiv L -E_1-E_2-E_3.\n$$\nSince\n$$\n(K_Y + 2 \\Delta_3 +(L -E_1-E_2-E_3)) \\Delta_3 = -2 < 0,\n$$\nby \\cite{burniat3}, lemma 5.1, we have\n\\begin{multline}\nH^0(Y, \\Omega^1_Y(\\log D_3)(K_Y+ \\mathcal{L}_2)) \\cong \\\\\n\\cong H^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4))(K_Y \n+ \\mathcal{L}_2 + \\Delta_3)).\n\\end{multline}\nNote that\n$$\nK_Y + \\mathcal{L}_3 + \\Delta_3 \\equiv S_1 + S_2  + E_2.\n$$\n\nAgain by lemma 5.1 in \\cite{burniat3} we see that\n\\begin{multline}\nH^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4))(K_Y + \n\\mathcal{L}_2 + \\Delta_3)) \\cong \\\\\n\\cong H^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4, \\log \nS_1, \\log S_2)(E_2)).\n\\end{multline}\nSince $E_2(f_1+f'_1+S_1 +S_2 + S_3 +S_4 + E_2) =  3$, we see by the \nsame arguments as in case 1) that\n\n\\begin{multline}\nh^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4, \\log S_1, \n\\log S_2)(E_2)) \\leq \\\\\n\\leq h^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4, \\log \nS_1, \\log S_2, \\log E_2)) + 2.\n\\end{multline}\n\n\\begin{claim}\nThe Chern classes of $f_1, S_1, \\ldots , S_4, E_2$ are linearly \nindependent.\n\\end{claim} \n\n\\begin{proof}[Proof of the claim.]\n\nAssume that \n\\begin{equation}\\label{zero3}\n\\lambda f_1 + a_1 S_1 + a_2 S_2 + a_3 S_3 +a_4 S_4 + \\mu E_2\\equiv 0,\n\\end{equation}\nwhere $\\lambda, a_1, a_2, a_3,a_4, \\mu \\in \\ensuremath{\\mathbb{C}}$.\n\nIntersection with $S_3, S_4$ gives $a_3 = a_4 =0$, whereas intersection with $E_1$ yields then  $a_1=0$.\n Intersection with $E_3$ instead gives  $a_2 =0$.\nFinally,  intersection with $E_4$ yields $\\lambda =0$, whence also $\\mu =0$. \\end{proof}\nTherefore\n$$\nh^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4, \\log S_1, \n\\log S_2, \\log E_2)) = 1,\n$$\nand we have shown 3).\n\n\n\\end{proof}\n\n\\begin{proof}[Proof of theorem \\ref{main}]\n(1) has been proved in theorem \\ref{homotopy}.\n\n(2) By \\cite{inoue}, page 318, $K_S$ is ample and by corollary \\ref{cortheta} the tangent space $H^1(S, \\Theta_S)$ to the base $\\Def(S)$ of the Kuranishi family of $S$ consists of the invariants for the action of the group $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^2$. Therefore all the local deformations of $S$ admit a $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^2$-action, hence are $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^2$ Galois coverings of the four nodal cubic surface (the anticanonical image of $Y$).\n\nFurthermore, the dimension of $H^1(S, \\Theta_S)$ is equal to the dimension of the Inoue  family containing $S$ in the moduli space $\\mathfrak{M}^{can}_{1,7}$, whence the base of the Kuranishi family of $S$ is smooth.\n\nSince the quotient of a smooth variety by a finite group (in our case, the automorphism group  $\\Aut(S)$) is normal, it follows that the irreducible connected component of the moduli space $\\mathfrak{M}^{can}_{1,7}$ corresponding to Inoue surfaces with $K_S^2=7, p_g = 0$ is normal and in particular generically smooth.\n\nThe family of Inoue surfaces is parametrized by a smooth (4-dimensional) rational variety (cf. e.g. remark \\ref{ratvar}), whence  unirationality follows.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAn elastic shell may be defined as a three-dimensional elastic body with one dimension that is much thinner than the other two \\citep{libai1998}. This enables us to consider the deformation of only the mid-plane of the elastic body in this thinner direction. A membrane may then be defined as a shell which has negligible resistance to bending \\citep{libai1998, humphrey2003}.  Elastic membranes are commonly found in biological and engineering contexts, where they may span relatively large areas despite having little volume or weight \\citep{humphrey1998}. \n\nThere are two different classes of non-linear elastic membrane theories and we refer to \\citet{haughton2001} for a more complete comparison of the two theories, but give a brief description here. The first of these membrane theories may be called a \\lq membrane-like shell\\rq \\citep{libai1998}, which takes the shell theory of three-dimensional elasticity and introduces the membrane assumption of no stress in the direction normal to the membrane. In this formulation the thickness is included in the derivation, as the principal stretch in the thickness direction, $\\lambda_n$, appears in the governing equations, which explicitly allows the membrane to get thinner to conserve mass when the constraint of incompressibility is imposed. This is a common method of treating membranes, and a comprehensive derivation may be found in \\citet{libai1998} and \\citet{steigmann2007}. A variational treatment may also be considered, for example see \\citet{ledret1996}.\n\nThe second class of membrane theories occurs from considering the mid-plane of the membrane as a two-dimensional sheet of elastic material embedded in  three-dimensional space, entirely neglecting thickness effects through the membrane \\citep{li1995}. \\citet{nadler2009} call this a \\lq simple membrane', and a consequence of this reduction is that the stretch through the membrane is not included in the formulation. The deformation gradient is then two-dimensional, and there are only two strain invariants and two principal stretches, rather than the three that arise in the three-dimensional theory,  see \\citet{steigmann2001,steigmann2009} for details.  \n\nIt has been shown that the simple membrane and membrane-like shell approaches give the same governing equations to leading order \\citep{naghdi1972, haughton2001, steigmann2007}, and it is the specification of the constitutive behaviour which varies between the two theories. It is possible to define two-dimensional strain-energy functions which have no counterpart in the three-dimensional theory, as discussed in \\citet{haughton2001}, as well as to use three-dimensional strain-energy functions in the two-dimensional theory. Further details of such matters may be found in \\citet{libai1998} and  \\citet{steigmann2007}.\n\nWhen considering deformations of membranes using either of the above theories, it is important to ensure that the membrane is in a state of tension, with both principal stresses remaining positive throughout the deformation \\citep{libai1998}. If compressive (negative) stresses occur then the membrane may wrinkle, a local buckling event \\citep{miyamura2000} where alternating crests and troughs appear parallel to the direction of principal tension \\citep{libai2002}, at which point additional refinements are required. In particular, \\citet{andra2000} show that equilibrium states with distributions of infinitesimal wrinkles are possible when bending stiffness is neglected, due to loss of convexity of the strain-energy function. \\citet{pipkin1986} showed how to modify the constitutive behaviour of the material in regions where compressive stresses exist, such that a pseudo-surface is considered which is in simple tension in these regions, neglecting the local details of the wrinkled structure in favour of global results; this is called tension field theory. For further details on treatments of wrinkling see for example, \\citet{pipkin1986}, \\citet{steigmann1990}, \\citet{epstein2001ani} and \\citet{libai2002}. A physical example of such wrinkles may be seen when a pointed object is pushed into a fixed curved piece of the plastic wrap, causing straight crests to radiate from the point of contact.\n\nThe indentation of an axisymmetric elastic membrane by a rigid body has been considered previously, although mostly through the use of linear elasticity. Here we are interested in large deformations, and therefore use non-linear elasticity theory to model the deflection of the membrane. Physical applications\/occurrences of such indentations include puncture of rubber gloves by medical needles \\citep{nguyen2009a, nguyen2009b}, stones embedding into rubber tyres \\citep{yang1971} and our specific interest of seed germination, which we shall discuss later.\n\n\\citet{yang1971} considered the indentation of a flat circular hyperelastic membrane by a rigid sphere, in the large deformation regime, using the three-dimensional Mooney-Rivlin strain-energy function. \\citet{liu2001} estimate the elastic modulus of a specific elastomeric film by comparing numerical simulations of the model introduced by \\citet{yang1971} with their experimental observations. \\citet{steigmann2005} and \\citet{nadler2006} consider the indentation and penetration of a material described by a compressible two-dimensional Varga strain-energy function by a spherical-tipped cylindrical indenter, and assume that the membrane will break if a punctured state, with a discontinuous membrane, is energetically favourable. \\citet{selvadurai2006} compares finite element simulations of indentation with a spherical indenter with experimental results, and provides a thorough literature review on rubber membranes, including a discussion of friction effects.\n\nThis rigid-body indentation is closely related to the adhesion behaviour of a rigid punch attached to a membrane, which exhibits stretching as the rigid body is pulled away. This has been considered by, for example, \\citet{nadler2008}, who show that non-linear elastic effects are important; they also find that the imposition of pre-stress on the material has a strong effect on the force required to separate the punch from the membrane.\n\n\\citet{haughton1996} consider the response of planar and hemispherical elastic membranes to filling with a specific volume of incompressible liquid, using a relaxed strain-energy function when the membrane becomes wrinkled. They find an instability whereby the depth of the liquid increases very rapidly with only a slight increase in volume. Depending on the strain-energy function this unstable behaviour may continue indefinitely or re-stabilise, corresponding to the bifurcation points of a uniformly inflated spherical membrane. \\citet{haughton1996} find that the plane membrane is a special case in that it remains tense throughout the deformation, whereas a hemispherical membrane contains a wrinkled section when the volume of liquid is small, which also reappears briefly after the instability.\n\nIn this article we shall generalise the previous work in three directions. Firstly, we allow the profile of the indenter tip to be of general axisymmetric shape rather than only considering spherical indenters, although we shall exclude sharply pointed tips by imposing mild continuity requirements. Secondly, we wish to investigate membranes which are initially curved, introducing this change of reference configuration into the elastic equilibrium equations. Lastly, we also wish to compare the effects of using different strain-energy functions to describe the constitutive behaviour of the membrane. \n\nOur primary motivation behind considering this deformation lies in seed germination, in particular the stretching and rupture of the endosperm by the root-tip in certain species of germinating seeds, specifically the two related Brassicaceae species \\textit{Arabidopsis thaliana} \\citep{holdsworth2008} and \\textit{Lepidium sativum} \\citep{muller2006}. These two species have approximately ellipsoidal seeds, including a thin, deformable tissue called the endosperm which surrounds the embryo. During the germination process the embryo grows, pushing the root-tip against the endosperm, stretching it before rupture occurs. The shape of the root-tip is approximately prolate spheroidal, as is the surrounding endosperm, hence our interest here in non-flat membranes and non-spherical indenter tips.\n\nThe endosperm itself is thin and highly curved, and rapidly dehydrates when separated from the rest of the seed. It is therefore difficult to use conventional techniques to measure the elastic properties of this tissue, and so an indirect method is required. We are particularly interested in determining the elastic properties of the endosperm as biological changes occur during germination which alter the cell wall microstructure.\n\nWe therefore wish to lay the mathematical groundwork before modelling biological puncture force experiments in a later work. In these experiments the seed is dissected and the embryo removed before a metal indenter is pushed through the endosperm, examples of such experiments are given in \\citet{muller2006}. The force and position of the indenter are measured during this experiment, and we then wish to estimate the elastic properties of the endosperm using this technique, particularly how these properties change over time. It has been shown in \\citet{muller2006} that the force required to puncture through \\textit{Lepidium sativum} endosperms decreases during the germination timescale. In preliminary experiments it has been shown that the vertical length of endosperm may double in size before rupture, clearly requiring non-linear elasticity to model this effect.\n\n\\section{Theory}\n\\subsection{Governing Equations}\nWe shall use the theory of non-linear elasticity as detailed in \\citet{ogden1997} to model the deformation. We assume that the midsurface of the undeformed membrane has symmetry about a particular axis, and may therefore be described by cylindrical coordinates with respect to the usual basis $(\\mathbf{e}_r,\\mathbf{e}_\\theta,\\mathbf{e}_z)$, \n\\begin{equation}\\label{undefcon}\n\\mathbf{X} = \\hat{R}(\\hat{S}) \\mathbf{e}_r(\\Theta) + \\hat{Z}(\\hat{S}) \\mathbf{e}_z, \\; 0 \\leq \\hat{S} \\leq \\hat{L}, \\; 0 \\leq \\Theta \\leq 2 \\pi, \n\\end{equation}\nwhere $\\hat{S}$ is the arclength along the reference midsurface, which will be used as the independent variable throughout. We shall take the origin of our coordinates to be the centre of the membrane, $\\hat{R}(0) = 0, \\hat{Z}(0) = 0$, as well as requiring that the membrane be smooth there, $\\hat{Z}'(0) = 0$.  If the membrane is initially flat we set $\\hat{R}(\\hat{S}) = \\hat{S}, \\hat{Z}(\\hat{S}) = 0$, in this case $\\hat{R}$ may be used as the independent coordinate as has been used in previous studies.\n\nWe are interested in axisymmetric deformations that map the reference configuration $\\mathbf{X}$ to the deformed configuration $\\mathbf{x}$, the midsurface of which is given by\n\\begin{equation}\\label{defcon}\n\\mathbf{x} = \\hat{r}(\\hat{S}) \\mathbf{e}_r(\\theta) + \\hat{z}(\\hat{S}) \\mathbf{e}_z, \\; 0 \\leq \\hat{S} \\leq \\hat{L}, \\; 0 \\leq \\theta \\leq 2 \\pi.\n\\end{equation}\nA point in the deformed surface is therefore given by two variables, $\\hat{r}$ and $\\hat{z}$, the three-dimensional shape being then given by rotation around the $z$ axis. Immediately, due to the imposed axisymmetry, we may identify $\\Theta = \\theta$. We also impose the same conditions as in the undeformed state at the axis, that is $\\hat{r}(0) = \\hat{z}(0)=0, \\hat{z}'(0)=0$.\n\nHere we shall solve the static problem, where we assume that there is no time-dependence involved in the deformation. This is appropriate to the quasi-steady cases in which we are interested, for which the indentation is slow compared to the inertia of the membrane.\n\nAppendix A outlines the derivation of the equilibrium equations for a curved  membrane, using the membrane-like shell approximation to the three-dimensional theory of elasticity, as discussed above. \nThe principal stretch ratios may be defined in the circumferential, radial, and normal directions in the form (see Appendix A for details),\n\\begin{equation}\\label{lambdas}\n\\lambda_s = \\dd{\\hat{s}}{\\hat{S}} = \\frac{\\sqrt{\\left(\\dd{\\hat{r}}{\\hat{S}}\\right)^2 + \\left(\\dd{\\hat{z}}{\\hat{S}}\\right)^2}}{\\sqrt{\\left(\\dd{\\hat{R}}{\\hat{S}}\\right)^2 + \\left(\\dd{\\hat{Z}}{\\hat{S}}\\right)^2}} , \\quad \\lambda_\\theta = \\frac{\\hat{r}(\\hat{S})}{\\hat{R}(\\hat{S})}, \\quad \\lambda_n = \\frac{\\hat{h}}{\\hat{H}}, \n\\end{equation}\nwhere $\\hat{h}$ and $\\hat{H}$ are the deformed and undeformed thicknesses and $\\hat{s}$ is the arclength in the deformed configuration. We assume that the membrane is made of a hyperelastic material with a strain-energy function given by  $\\hat{W}(\\lambda_s,\\lambda_\\theta,\\lambda_s^{-1} \\lambda_\\theta^{-1})$, where we have already assumed that the material is incompressible and therefore set $\\lambda_s \\lambda_\\theta \\lambda_n = 1$; see \\citet{ogden1997} for more details. The principal stress resultants per unit length in the deformed membrane are therefore defined as\n\\begin{equation}\\label{stressres}\n\\hat{T}_s = \\frac{\\hat{H}}{\\lambda_\\theta} \\pafrac{\\hat{W}}{\\lambda_s}, \\quad\n\\hat{T}_\\theta = \\frac{\\hat{H}}{\\lambda_s} \\pafrac{\\hat{W}}{\\lambda_\\theta},\n\\end{equation}\nwhich may be seen to be $\\hat{T}_\\alpha = \\hat{h} \\hat{\\sigma}_\\alpha$, where $\\hat{\\sigma}_\\alpha = \\lambda_\\alpha \\pafrac{\\hat{W}}{\\lambda_\\alpha}$ is the usual principal Cauchy stress in incompressible three-dimensional elasticity, and similarly for $\\hat{T}_\\theta$. Appendix A shows how the stresses only occur in the combination $\\hat{h}\\hat{\\sigma}_\\alpha$, and hence we introduce $\\hat{T}_\\alpha$ with dimensions of force per length.\n\nWhile the theory presented here is appropriate for any isotropic incompressible strain-energy function, we shall study some specific examples in order to make later comparisons. One commonly used such function is the Mooney-Rivlin strain-energy function,\n\\begin{equation}\\label{mooney}\n\\hat{W} = \\frac{\\mu}{2(1+\\alpha)}\\left[(1-\\alpha) \\left( \\lambda_s^2 + \\lambda_\\theta^2 + \\lambda_s^{-2} \\lambda_\\theta^{-2} - 3 \\right) + \\alpha \\left( \\lambda_s^{-2} + \\lambda_\\theta^{-2} + \\lambda_s^{2} \\lambda_\\theta^{2} - 3 \\right) \\right],\n\\end{equation}\nwhere $\\mu$ is the infinitesimal shear modulus of the elastic material and $0\\leq \\alpha \\leq 1$ is a parameter that controls the deviation from a Hookean response. An important special case of \\rr{mooney} is the neo-Hookean strain-energy function, which occurs when $\\alpha=0$. The $(1+\\alpha)$ in the denominator of \\rr{mooney} is necessary to ensure that $\\mu$ is the infinitesimal shear modulus of the material. \\citet{fung1993} introduced a strain-energy function designed for the modelling of biological soft tissues such as healthy arterial tissue, which commonly feature strongly  strain-stiffening behaviour,\n\\begin{equation}\n\\hat{W} = \\frac{\\mu \\left(e^{\\Gamma \\left(\\lambda_s^2 + \\lambda_\\theta^2 + \\lambda_s^{-1}\\lambda_\\theta^{-1} -3\\right)} - 1\\right)}{2 \\Gamma} \\label{Fung},\n\\end{equation}\nwhere $\\Gamma$ is a positive parameter representing the degree of strain stiffening. In the limit as $\\Gamma$ approaches zero, \\rr{Fung} also recovers the neo-Hookean strain-energy function.\n\nAs shown in Appendix A, the equilibrium equations in the tangential and normal directions are respectively given by,\n\\begin{subequations}\\label{equib}\n\\begin{gather}\n(\\hat{r} \\hat{T}_s)' -\\hat{r}' \\hat{T}_\\theta =0, \\label{equib1}\\\\\n\\hat{\\kappa}_s \\hat{T}_s + \\hat{\\kappa}_\\theta \\hat{T}_\\theta = \\hat{P},\\label{equib2}\n\\end{gather}\n\\end{subequations}\nwhere $\\hat{P}(S)$ represents the pressure difference across the membrane in the normal direction, and the principal curvatures $\\hat{\\kappa}_s,\\hat{\\kappa}_\\theta$ are defined in \\rr{curvatures}. The system \\rr{equib} contains terms involving $\\hat{r}, \\hat{r}',\\hat{r}'',\\hat{z}',\\hat{z}''$, and is therefore a third order differential system, as $\\hat{z}$ does not appear explicitly. Rearranging \\rr{equib1} for $\\hat{T}_\\theta$ and using Codazzi's equation, \\rr{codazzi2}, we may integrate \\rr{equib2} to find,\n\\begin{equation}\\label{intequib}\n\\hat{r}^2 \\hat{\\kappa}_\\theta \\hat{T}_s = \\frac{\\hat{F}}{2 \\pi} + \\frac{\\hat{P} \\hat{r}^2}{2} - \\frac{1}{2} \\int \\dd{\\hat{P}}{\\hat{S}} \\hat{r}^2 d \\hat{S},\n\\end{equation}\nwhich may be used, provided $\\hat{P}$ is constant, to reduce the differential order of the system by one. We note that this integral exists for a membrane of any undeformed shape, and corresponds to the resultant in the $\\hat{Z}$ direction. The integration constant $\\hat{F}$ represents the total transverse force acting on the membrane \\citep{nadler2006}.\n\n\\subsection{Wrinkling}\nUsing \\rr{equib} we may calculate the deformed shape of the membrane, assuming that the stress resultants both stay positive over the region. If either stress resultant becomes negative then the theory as outlined above is not appropriate, as a true membrane can not support compressive stresses. The regions with such a negative compressive stress are assumed to have wrinkled, with the crests of the wrinkles perpendicular to the compressive stress direction; this is a local buckling instability \\citep{jenkins1991}. We therefore introduce tension-field theory to consider a smoothed-out \\lq pseudo-surface' that is in equilibrium, with the corresponding stress exactly zero everywhere. \n\nFor the strain-energy functions considered here, the minimum of $W$ with respect to $\\lambda_\\theta$, subject to both stretches being positive, occurs  when $\\lambda_\\theta = \\lambda_s^{-1\/2} \\equiv w(\\lambda_s)$, with an equivalent expression for $\\lambda_s$. This is termed the natural width of the strip in tension by \\citet{pipkin1986}. Following \\citet{pipkin1986} we introduce the tension-field theory by defining a relaxed strain-energy density function as,\n\\begin{equation}\nW^{*}(\\lambda_s,\\lambda_\\theta) =  \n\\left\\{\n\t\\begin{array}{lcc}\n\\hat{W}(\\lambda_s,\\lambda_\\theta) & \\lambda_s \\geq 1, & \\lambda_\\theta \\geq 1 \\\\\n\\hat{W}(w(\\lambda_\\theta),\\lambda_\\theta) & \\lambda_s \\leq w(\\lambda_\\theta), & \\lambda_\\theta>1 \\\\\n\\hat{W}(\\lambda_s,w(\\lambda_s)) & \\lambda_s>1, & \\lambda_\\theta \\leq w(\\lambda_s) \\\\ 0  &\\lambda_s<1, & \\lambda_\\theta<1\n\\end{array}\n\\right.\n\\end{equation}\nwhich is the quasiconvexification of $\\hat{W}$. Using this definition of $W^{*}$ we recover the normal behaviour in the taut region while resolving the wrinkled regions. When both stretches are less than unity, the membrane becomes slack, with no tension in either direction. If this occurs here then \\rr{intequib} implies that $\\hat{F}=0$ and thus there is no indentation, when $\\hat{P}=0$.\n\n\nWe find numerically that the wrinkled region, if it exists, occurs when $\\hat{T}_\\theta = 0$. We therefore have the following system of equations in the wrinkled region,\n\\begin{equation}\\label{equibwrink}\n\\hat{\\kappa}_s \\hat{T}_s = \\hat{P}, \\;\\; \\left(\\hat{r} \\lambda_s^{1\/2} \\hat{W}_1(\\lambda_s,\\lambda_s^{-1\/2})\\right)'=0.\n\\end{equation}\nWhen the pressure $\\hat{P}$ is zero, \\rr{equibwrink}$_1$ implies that either the membrane is slack (and thus there is no indentation) or $\\kappa_s =0$. We may therefore integrate \\rr{equibwrink} to give,\n\\begin{equation}\\label{equibwrink2}\n\\frac{\\hat{z}'}{\\hat{r} \\sqrt{\\hat{r}'^2 + \\hat{z}'^2}} = \\text{const.}, \\;\\; \\hat{r} \\lambda_s^{1\/2} \\hat{W}_1(\\lambda_s,\\lambda_s^{-1\/2}) = \\text{const.},\n\\end{equation}\nwhere the constants are determined by the values of $\\hat{r}$ and $\\hat{z}$ at the point at which the membrane first becomes wrinkled, $\\hat{S}=\\hat{S}_1$. \n\nThe first order differential equation in \\rr{equibwrink2} may then be integrated until $\\tilde{\\lambda}_\\theta = \\lambda_\\theta$, at which point $\\hat{T}_\\theta$ given by \\rr{stressres} becomes zero again, this is $\\hat{S}=\\hat{S}_2$. This process may then be repeated if the membrane became wrinkled again, if required. Clearly, in the pseudo-surface the lines in the $s$-direction must be straight as the associated curvature $\\hat{\\kappa}_s$ is zero. \\citet{libai2002} present a general algorithm for dealing with axisymmetric wrinkling, and this approach conforms to their technique.\n\n\\citet{haughton1995} and \\citet{haughton1996} state that the extent (in terms of $\\hat{S}$) of the wrinkled region is determined solely by the tense theory, and suggest that if you use the tense theory even when $\\hat{T}_\\theta <0$, then $\\hat{T}_\\theta = 0$ at $\\hat{S}_2$. However, this is contrary to what we find numerically here, and so we integrate the wrinkled region exactly.\n\n\\subsection{Indenter Profile}\nWe assume that the indenter is a cylinder with a curved tip, with a specified axisymmetric profile where the components in the $\\mathbf{e}_r$ and $\\mathbf{e}_z$ directions are given parametrically in terms of an angle $\\phi$ from the negative $z$-axis. We therefore assume the indenter surface is described by \n\\begin{gather}\\label{indenter}\n\\psi(\\phi) = \\left\\{ \\begin{array}{lc}\n \\hat{\\rho} A(\\phi) \\mathbf{e}_r + \\hat{\\rho} B(\\phi) \\mathbf{e}_z, &  0 \\leq \\phi \\leq \\pi\/2, \\\\\n \\hat{\\rho} \\mathbf{e}_r + (\\hat{\\rho} B(\\pi\/2) + \\hat{\\rho} \\phi)\\mathbf{e}_z,  & \\phi >\\pi\/2 \n\\end{array}\n\\right.\n\\end{gather}\nwhere $\\hat{\\rho}$ is the radius of the cylindrical part of the indenter to which this curved tip is attached. We require that the axisymmetry around the $z$ axis is maintained and hence $A(0)=0$, and also that the indenter tip smoothly connects to the cylindrical part, $A(\\pi\/2) = 1, A'(\\pi\/2)=0$. In addition, we require the smoothness condition of $B'(0)=0$, which prevents the consideration of perfectly sharp pointed tips, although we may approach these. We shall also impose the restriction that the indenter profile is convex, although for some non-convex profiles replacing the profile by its convex hull may give the appropriate shape, although the calculation of the load exerted on the membrane may need to be adjusted.\n\nIf the indenter tip is spherical, then $A(\\phi) = \\sin \\phi, B(\\phi) = - \\cos \\phi$, where the negative sign merely ensures that we consider the  indentation to occur in the negative $\\mathbf{e}_z$ direction. Spheroidal tips may be considered by setting $B(\\phi) = - \\gamma \\cos \\phi$, where $\\gamma > 1$ for a prolate spheroid and $0<\\gamma < 1$ for an oblate spheroid. As $\\gamma \\to 0$, the case of the indentation by a flat cylindrical punch is approached, while $\\gamma \\to \\infty$ approximates indentation by a sharp tip. \n\nWe note that it is possible to use an indenter which consists of just the \\lq tip' by modifying \\rr{indenter}, for instance when a solid sphere is allowed to deform the membrane under gravity as in the experiments of \\citet{liu2001}. For a given applied force this may be modelled by a spherical-tipped cylindrical indenter, provided that the deformed surface does not touch the sides of the cylinder. Indeed, it has been stated by \\citet{selvadurai2006} and \\citet{nadler2006} that the deformed surface never makes contact with the cylindrical part prior to membrane penetration, although our preliminary experimental results and the second figure in \\citet{nguyen2004mech} suggest otherwise, so perhaps that statement is a specification to their equations. In particular for the Varga strain-energy function we agree with \\citet{nadler2006}, but not for general strain-energy functions. Hence we do allow the membrane to touch the cylindrical part of the indenter, where we set $\\hat{r}(\\hat{S}) \\equiv \\hat{\\rho}$ and solve for $\\hat{z}$, although these regions will be dashed on the load-indentation graphs presented in Section \\ref{results}.\n\n\\subsection{Solution Procedure}\nWe now non-dimensionalise with respect to the undeformed radius, $\\hat{R}_L= \\hat{R}(\\hat{L})$, and the infinitesimal shear modulus, $\\mu$,\n\\begin{gather}\n(\\hat{S},\\hat{R},\\hat{Z},\\hat{H},\\hat{r},\\hat{z}, \\hat{\\rho},\\hat{L}) = \\hat{R}_L (S,R,Z,H,r,z,\\rho,L), \\;  (\\hat{T}_s, \\hat{T}_\\theta) = \\mu \\nonumber\n\\hat{R}_L H (T_s,T_\\theta), \\\\\n\\hat{P} = \\mu P, \\; \\hat{F} =  \\mu \\hat{R}_L^2 H F, \\; \\hat{W} = \\mu W.\n\\end{gather}\nFor flat membranes $\\hat{R}_L \\equiv \\hat{L}$, but for a curved membrane this is not generally true. This rescaling leaves the previously stated equations for the stretches, stress resultants and curvatures unchanged, apart from dropping the hats from all variables and removing $\\hat{H}$ from \\rr{stressres}. The parameter $\\rho$ is now a non-dimensional ratio relating the radius of the indenter to the radius of the undeformed membrane.\n\nThe problem may be divided into two distinct regions of $S$, that is the region in the undeformed geometry that is now in contact with the indenter, $0 \\leq S \\leq S_c$, and the outer region where the membrane is free, $S_c < S \\leq L$. For a flat membrane, $S_c$ is the radius of the circle in the undeformed membrane which is in contact with the indenter in the deformed configuration, while for a curved reference surface $R(S_c)$ is the radius of the corresponding circle in the undeformed membrane. If wrinkling occurs in  the free region there will be four distinct regions instead, as we find numerically that the membrane returns to biaxial tension by $S=L$ in the examples considered here.\n\nWhile the membrane is in contact with the indenter, we assume that the deformed shape follows the surface of the indenter and we therefore have,\n\\begin{equation}\nr(S) = \\rho A(\\phi(S)), \\; z(S) = - \\delta + \\rho B(\\phi(S)),\n\\end{equation}\nwhere $\\delta$ is the depth of indentation to be found as part of the solution. Therefore in the contact region,\n\\begin{equation}\\label{indentstretch}\n\\lambda_s = \\rho \\frac{\\sqrt{\\left(\\dd{A}{\\phi}\\right)^2 + \\left(\\dd{B}{\\phi}\\right)^2}}{\\sqrt{R'^2 + Z'^2}} \\dd{\\phi}{S}, \\; \\lambda_\\theta = \\rho \\frac{A(\\phi(S))}{R(S)},\n\\end{equation}\nwhere we have used $\\phi'(S) >0$. We may therefore evaluate the first equilibrium equation, \\rr{equib1}, to find $\\phi(S)$ given appropriate boundary conditions. The second equilibrium equation enables us to calculate the pressure $P$ exerted by the indenter on the membrane after calculating the membrane deformation, but is auxiliary to computing the deformation itself in the contact region.\n\n\\subsection{Boundary Conditions}\nWe now specify the boundary conditions in order to integrate the equations above. The requirement of axisymmetry implies that $\\phi(0)=0$, while using the restrictions on $A$ and $B$ we find $\\phi'(0) = \\rho^{-1} \\lambda_s(0)\/A'(0)$, and we can therefore integrate \\rr{equib1} to find $\\theta(S)$ for a given stretch at the pole $\\lambda_s(0)$.\n\nIn the region $S_c<S \\leq L$, the membrane is governed by \\rr{equib} with $P=0$, which gives us a third order system for $r$ and $z$, or $\\lambda_s$ and $\\lambda_\\theta$. We require continuity of the membrane shape, and so we prescribe $r(S_c),r'(S_c),z'(S_c)$ to be those given from the inner region.\nWe allow for the membrane to be stretched prior to indentation, imposing a pre-stretch of $\\lambda_p \\geq 1$ on the material, that is $\\lambda_p = r(L)$.\nThis pre-stretch corresponds to moving the circular boundary $S=L$ outwards in the $\\mathbf{e}_r$ direction, inducing a deformation in the membrane; if the membrane is not initially flat then this will have the result of changing the shape of the initial profile by decreasing the vertical \\lq height' of the membrane, as the material moves towards the level of the boundary circle.\n\nUtilising a shooting method we specify the stretch at the pole, $\\lambda_s(0)=\\lambda_\\theta(0) = \\lambda_0$, and then guess a value of $S_c$. We then iterate on $S_c$ to satisfy the condition $\\lambda_\\theta(L) =  \\lambda_p$. Having found the appropriate value of $S_c$ for a given $\\lambda_0$ and $\\lambda_p$, we may then calculate the value of the central displacement of the membrane which is given by,\n\\begin{equation}\\label{zint}\nz_h=\\int_0^L \\dd{z}{S} \\; dS  = \\int_0^{S_c} \\dd{z}{S} \\;dS + \\int_{S_c}^L \\dd{z}{S} \\; dS,\n\\end{equation} \nwhere the solutions in the appropriate regions are substituted into each integral of \\rr{zint}. From this we calculate the indentation depth, $\\delta = z_h - Z_h$, where $Z_h = Z(L)-Z(0)$ is the original depth of the membrane. Having done this, we may vary $\\lambda_0$ and repeat the process in order to solve for a variety of indentation depths, or equivalently for values of the applied force $F$.\n\n\nWe therefore require the following parameters to be specified: the membrane prestretch $\\lambda_p$, the radius of the cylindrical part of the indenter $\\rho$, the undeformed membrane shape $(R,Z)$ and the shape of the indenter tip $(A,B)$. In addition, the strain-energy function must be specified, along with the value of any parameters included therein. The choice of strain-energy function is not a trivial matter in the large indentation regime, and we will show how the force required to indent the membrane is strongly dependent on the form of the strain-energy function used, past the initial linear regime.\n\n\\section{Results and Discussion}\\label{results}\n\\begin{figure}[!htbf]\n \\psfrag{aa}{$r$}\n \\psfrag{bb}{$z$}\n\\centering \\epsfig{figure=indent1.eps,width=.6\\textwidth}\n\\caption{Deformed configuration after indentation of a flat membrane by a spherical indenter for, from top to bottom, $\\lambda_0 = 1.05,1.1,1.2,1.4,1.6,2$, with $\\lambda_p = 1$ and $\\rho = 0.2$. The strain-energy function used is the Mooney Rivlin with $\\alpha=0.1$.}\n\\label{sphereindent}\n\\end{figure}\nFigure \\ref{sphereindent} shows how the deformed configuration changes with increasing $\\lambda_0$ for a spherical indenter and a flat membrane, reproducing Figure 2 of \\citet{yang1971}. It may be seen that the deformed shape goes from having positive curvature while in contact with the indenter to a negative curvature in the free region. This is as expected as equation \\rr{equib2} implies that an odd number of the quantities $T_s,T_\\theta,\\kappa_s,\\kappa_\\theta$ must be negative in the free region, and therefore in unwrinkled parts of the membrane one of the curvatures must be negative. When the membrane is initially flat wrinkling is never observed, as found by \\citet{haughton1995} for the fluid-loaded case.\n\n\n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$}\n\\centering \\epsfig{figure=varyshapecolour2.eps,width=.6\\textwidth}\n\\caption{\n a variety of spheroidal indenter tips with $A(\\theta) = \\sin \\theta, B(\\theta) = - \\gamma \\cos \\theta, \\lambda_p=1,\\rho=0.2$. From top to bottom, $\\gamma = 0.005,0.125,0.25,0.5,1,2,4,6,8,200$}\n\\label{varyshapecolour}\\label{dispindent}\n\\end{figure}\n\n\nHaving calculated the deformed shape of the membrane we may evaluate the transverse force $F$ in the free region, this corresponds to an indentation displacement $\\delta$. Figure \\ref{dispindent} shows a variety of non-dimensional force-indentation curves for spheroidal indenter tips of varying shape. As is intuitively obvious, blunter indenter tips require more force to achieve the same indentation depth. The two extremal curves in Figure \\ref{dispindent} approach indentation with a flat punch and a point load respectively. In Figure \\ref{dispindent}, and following figures, dashed lines represent where the membrane has come into contact with the cylindrical part of the indenter, which occurs sooner with flatter tips.\n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$}\n\\psfrag{lab1}{\\tiny neo-Hookean}\n\\psfrag{lab2}{\\tiny Mooney-Rivlin, $\\alpha=0.1$}\n\\psfrag{lab3}{\\tiny Mooney-Rivlin, $\\alpha=0.5$}\n\\psfrag{lab4}{\\tiny Fung, $\\Gamma = 1$}\n\\centering \\epsfig{figure=indent3.eps,width=.8\\textwidth}\n\\caption{Force-indentation curves for the indentation by a spherical indenter of a flat membrane modelled using the neo-Hookean, Mooney-Rivlin and Fung strain-energy functions with $\\lambda_p=1,\\rho=0.2$. }\n\\label{sevary}\n\\end{figure}\n\nFigure \\ref{sevary} shows the load-indentation curves for a variety of strain-energy functions, where it may be seen that for this set of parameters up to $\\delta \\approx 0.4$ the material responses are identical, which is consistent to the results of \\citet{pujara1978} regarding the inflation of circular membranes. Past this range the varied stress-strain behaviour of the strain-energy functions give different loads for the same indentation, and thereafter the specification of the strain-energy function becomes important. The Fung material model, generally used for biological tissues such as arteries and muscles, predicts much higher loads than the other models for the same amount of indentation, as expected due to the strain-stiffening behaviour the Fung model encapsulates.\n\nThe neo-Hookean material model gives a horizontal asymptote in Figure \\ref{sevary}, with a constant $F$ corresponding to an infinite range of indentation. This is a consequence of using the neo-Hookean strain-energy function outside its range of validity.\nThe Mooney-Rivlin material gives linear growth of $F$ with $\\delta$, depending on the value of $\\alpha$, while the Fung material shows an exponential growth.\n\nIn particular, when the membrane is touching the cylindrical portion of the indenter we have $r' = 0$, and therefore evaluating \\rr{intequib} at $S=S_c$ gives,\n\\begin{equation}\\label{Fcyl}\nF = \\left. 2 \\pi R(S_c) \\pafrac{W}{\\lambda_s} \\right|_{S=S_c},\n\\end{equation}\nwhich explains how the response is dominated by the behaviour of the stress-strain relationship, as \\rr{Fcyl} indicates that force is proportional to the stress.\n\n\\begin{figure}[!htbf] \n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$}\n\\centering \\epsfig{figure=flatvaryrho.eps,width=.6\\textwidth}\n\\caption{Force-indentation curves for the indentation of a flat Mooney-Rivlin membrane by a spherical indenter of varying size, from bottom to top, $\\rho=0.1,0.2,\\ldots,0.9$}\n\\label{rhovary}\n\\end{figure}\nFigure \\ref{rhovary} shows how the size of the indenter affects the load behaviour, although the quantitative response behaviour is not affected. For large indentation depths the curves are parallel to each other, both before and after the membrane begins to touch the sides of the indenter. \n\nIn all the force-indentation curves for flat membranes presented above, in the linear elastic limit the relation $F \\propto \\delta^3$ may be seen, corresponding to the Schwerin-type solution in linear elasticity \\citep{komaragiri2005}. \\citet{begley2004} derive an approximate solution with such a cubic relationship for a finite-sized spherical indenter and a flat membrane, but we do not agree numerically with their proportionality constant. \n\n\\begin{figure}[!htbf] \n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$}\\centering \\epsfig{figure=linearlimitsph.eps,width=.6\\textwidth}\n\\caption{Close-up of the force-indentation curve for a spherical indenter and a flat membrane with the Mooney Rivlin strain-energy function with $\\alpha=0.1$, $\\lambda_p=1$, $\\rho=0.2$. Curve is given by $F=4.7292 \\delta^3$.}\n\\label{sphcubic}\n\\end{figure}\n\n\\begin{figure}[!htbf] \n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$}\n\\centering \\epsfig{figure=initspher.eps,width=.6\\textwidth}\n\\caption{Deformation of an initially spherical surface $R(S) = \\sin S, Z(S) = - \\cos S, \\rho = 0.4, \\lambda_p=1$ with a spherical indenter and a Mooney-Rivlin strain-energy function with $\\alpha=0.1$. From top to bottom: $\\lambda_0 = 1.2,1.4,1.6,2$. Dashed line shows the undeformed shape of the membrane, dotted sections of the deformed surface are wrinkled.}\n\\label{initspher}\n\\end{figure}\n\nFigure \\ref{initspher} shows the indentation of an initially spherical membrane by a spherical indenter of smaller radius. It can be seen that the deformed shape does not follow the curvature of the reference surface, which may be expected from the previous comment that $\\kappa_s$ or $T_\\theta$ must be negative, and from the fact that the governing equations do not include a term involving the reference curvatures, i.e. \\rr{curvatures} but with $r$ and $z$ replaced by $R$ and $Z$. In order to include such a term in the governing equations, we would need to consider a shell theory where the energy depends on the curvature of the shell in addition to the stretching, as discussed in \\citet{steigmann2001} for example. This increases the order of the governing equations from third to seventh order, and also introduces the question of determining how the strain-energy function depends on a further set of invariants; an additional two which depend only on the curvatures and three coupling terms between the curvatures and the stretches.\n\n \n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$} \\psfrag{lab1}{$\\zeta = 0$}\n\\psfrag{lab2}{$\\zeta = 0.125$}\n\\psfrag{lab3}{$\\zeta = 0.25$}\n\\psfrag{lab4}{$\\zeta = 0.5$}\n\\centering \\epsfig{figure=varyindentcurved.eps,width=.6\\textwidth}\n\\caption{Load-indentation curves for the indentation of an initially curved membrane with $R(S) = S, Z(S) = -\\cos (\\pi S\/2)\/4$ by various spheroidal indenter tips, from top to bottom: $\\gamma = 0.25,0.5,1,2,4,8$ with $\\lambda_p=1,\\rho=0.2$}\n\\label{RZvary}\n\\end{figure}\nFigure \\ref{varyindent2} shows the load-indentation curves for initially curved surfaces of the form, $R(S) = S, Z(S) = - \\psi \\cos ( \\pi S \/2), 0 \\leq S \\leq 1$. As $\\psi \\to 0$ we recover the flat membrane, and we see that as the reference surface departs from this shape the required load for greater indentation increases. \n\nFor pronouncedly curved reference surfaces, $\\psi \\approx 0.5$, we find that the theory used here gives a vanishingly small load for a finite indentation displacement. As we take the numerical limit $\\lambda_0 \\to 1$, the calculated load tends towards zero but there is a non-zero displacement, $\\delta \\approx 0.09$ in the membrane shown in Figure \\ref{varyindent2}. We believe that the reason this occurs is that the membrane wrinkles in the indentation region, where we are prescribing the displacement to be that of the indenter.\n\n\nThis limitation does not prevent us from solving for larger values of $\\lambda_0$ due to the quasi-static solution procedure used here. Figure \\ref{varyindent3} shows the same curved membrane as Figure \\ref{varyindent2}, but with the Fung strain-energy function instead of the Mooney-Rivlin. The linear elastic behaviour of the two graphs is the same, but at large displacement or load the impact of the strain-energy function may be seen. \n\n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$} \n\\centering \\epsfig{figure=varycurvedxx.eps,width=.6\\textwidth}\n\\caption{Load-indentation curves for the indentation with a spherical indenter of an range of initially curved membranes with $R(S) = 2 S\/ \\pi, Z(S) = -\\psi \\cos S$, from top to bottom: $\\psi=0.5,0.25,0.125,0.1,0.05,0$ and $\\lambda_p=1,\\rho=0.2$.}\n\\label{varyindent2}\n\\end{figure}\n\n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$} \n\\centering \\epsfig{figure=varycurvedxxFung.eps,width=.6\\textwidth}\n\\caption{Load-indentation curves for the indentation with a spherical indenter of an range of initially curved membranes with $R(S) = 2 S\/ \\pi, Z(S) = -\\psi \\cos S$, from top to bottom: $\\psi=0.5,0.25,0.125,0.1,0.05,0$ and $\\lambda_p=1,\\rho=0.2$, using the Fung strain-energy function.}\n\\label{varyindent3}\n\\end{figure}\n\n\n\nWhen the membrane is flat, both the stress resultants stay positive and wrinkling does not occur. For curved membranes, it is possible to increase the radial pre-stretch $\\lambda_p$ or the indentation distance $\\delta$ to ensure that the membrane remains in tension in the deformed configuration. This finding agrees directly with that of \\citet{haughton1996} for fluid filled membranes. For instance, if a spherical indenter nearly fills a spherical membrane, i.e. $\\rho$ is close to one, then wrinkling is only found for very small levels of $\\lambda_0$ with no pre-stretch, and eliminated entirely for slight amounts of pre-stretch, such as $\\lambda_p \\geq 1.1$. However, the imposition of the pre-strain will change the shape of the non-flat membrane prior to indentation, requiring knowledge of the shape of the membrane prior to the pre-strain, which may not be available in experimental situations.\n\n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\delta$}\n\\psfrag{bb}{$S_c$}\n\\psfrag{lab1}{\\tiny NH}\n\\psfrag{lab2}{\\tiny MR, $\\alpha=0.1$}\n\\psfrag{lab3}{\\tiny MR, $\\alpha=0.5$}\n\\psfrag{lab4}{\\tiny Fung, $\\Gamma = 1$}\\centering \\epsfig{figure=scvary.eps,width=.6\\textwidth}\n\\caption{Position of contact point $S_c$ as indentation depth changes for a indentation of a flat membrane with a spherical indenter, $\\lambda_p=1, \\rho=0.2$. The neo-Hookean (NH), Mooney-Rivlin (MR) and Fung strain-energy functions are shown.}\n\\label{scvary}\n\\end{figure}\n\nAs the indenter is pushed into a flat membrane the contact point $S_c$ initially increases as more of the indenter comes into contact with the material, before reaching a maximum value and then decreasing, as shown in Figure \\ref{scvary}. This later decrease is due to the fact that, at larger indentation depth, a comparatively small area of the reference membrane is highly stretched over the indenter tip. Once the membrane begins to touch the cylindrical sides of the indenter the behaviour of $S_c$ is governed by the strain-energy function in the same manner as $F$. This behaviour is found by \\citet{nadler2006} for the two-dimensional compressible Varga strain-energy function, although in this case the membrane never comes into contact with the cylindrical part of the indenter. \n\n\\section{Conclusions}\nWe have shown how to extend the previously presented non-linear elasticity theories of indentation with a finite radius indenter to include the effects of different strain-energy functions, non-spherical indenter tips and initially curved membranes. The importance of  correctly specifying the strain-energy function is shown clearly, along with the effects that changing the shape of the indenter has. \n\nIndentation tests are a useful method of determining the properties of an elastic membrane at large strain, and with experimental data from a wide variety of shapes and sizes of indenter a suitable strain-energy function may be fitted to the data, giving more information than merely spherical indenter tips. One difficulty may be in identifying where the start of the force-indentation curve occurs experimentally, due to the detection limits of the equipment used, which may mean that the beginning of the curve is missed.\n\n\nAs mentioned in the introduction, we plan to compare with experimental data from puncture force experiments on germinating seeds, with the aim of estimating the elastic properties of the endosperm using the theory presented here. The experiments will result in load-indentation curves, which we may then use to fit to the load-indentation curves resulting from the method here. In this case, the unknown parameters are the infinitesimal shear modulus and the strain-energy function of the material, and we shall attempt to characterize these through the  indirect indentation method.\n\n\\section*{Acknowledgements}\nThis work has been funded by an ERANET-PG grant for the vSEED project (BBSRC grant BBG02488X1). JRK additionally acknowledges the support of the \nRoyal Society and the Wolfson Foundation.\n\n\\section*{Appendix}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\\setcounter{equation}{0}\n\nIn this appendix we shall provide a sketch of the derivation of the equilibrium equations given in \\rr{equib}, using the membrane-like shell approach. For full details including proofs and error estimates of the derivation here see \\citet{haughton2001} or \\citet{haughton1978a}. Where possible we shall use upper case letters to describe the undeformed configuration and lower case letters to describe the deformed configuration. In addition, Greek indices vary over $(1,2)$, and the summation convention for repeated indices is used unless otherwise noted.\n\nWe consider a thin shell with position vector,\n\\begin{equation}\n\\mathbf{X} = \\mathbf{Y}(\\xi^1, \\xi^2) + \\xi^3 \\mathbf{N}(\\xi^1,\\xi^2),\n\\end{equation}\nwhere $(\\xi^1,\\xi^2)$ are curvilinear coordinates describing the midsurface of the shell, $-H\/2 \\leq \\xi^3 \\leq H\/2$ is a thickness coordinate in the positive normal direction $\\mathbf{N}$ and $H$ is the thickness of the undeformed shell. We assume that the shell is thin enough such that if $\\eta$ is the minimum principal radii of curvature of the midsurface then,\n\\begin{equation}\n\\epsilon = \\frac{H_{\\textrm{max}}}{\\eta} \\ll 1,\n\\end{equation}\nwhere $H_{\\textrm{max}}$ is the maximum thickness throughout the shell.\n\nWe assume that the position vector $\\mathbf{x}$ of the deformed membrane may be decomposed in the same way into a surface vector $\\mathbf{y}$ and a thickness component in the deformed normal direction $\\mathbf{n}$,\n\\begin{equation}\n\\mathbf{x} = \\mathbf{y}(\\zeta^1,\\zeta^2) + \\zeta^3 \\mathbf{n}(\\zeta^1,\\zeta^2),\n\\end{equation}\nwhere $(\\zeta^1,\\zeta^2)$ are coordinates describing the deformed surface of the shell and $-h\/2 \\leq \\zeta^3 \\leq h\/2$ expresses the distance through the thickness direction of the membrane. \\citet{haughton2001} show that, correct to within $\\mathcal{O}(\\epsilon)$, the surface $\\zeta^3=0$ corresponds to the midsurface of the reference membrane $\\xi^3=0$, so that $\\mathbf{y}$ may be used as the midsurface of the deformed membrane. Expanding $\\zeta^3$ around $\\xi^3$, we may also find that $h= H \\zeta^3_{,3}(\\xi^1,\\xi^2,0)$ to the same order. The tangent vectors to the undeformed and deformed surfaces may be written as,\n\\begin{equation}\n\\mathbf{G}_\\alpha = \\mathbf{Y}_{,\\alpha}, \\; \\mathbf{g}_\\alpha = \\mathbf{y}_{,\\alpha},\n\\end{equation}\nand we also define dual vectors,\n\\begin{equation}\n\\mathbf{G}^\\alpha = G^{\\alpha \\beta} \\mathbf{G}_\\beta, \\; \\mathbf{g}^\\alpha = g^{\\alpha \\beta} \\mathbf{g}_\\beta,\n\\end{equation}\nwhere $G^{\\alpha \\beta} G_{\\alpha \\beta} = \\delta^{\\alpha}_{\\beta}, G_{\\alpha \\beta} = \\mathbf{G}_\\alpha \\cdot \\mathbf{G}_\\beta, g^{\\alpha \\beta} g_{\\alpha \\beta} = \\delta^{\\alpha}_{\\beta}, g_{\\alpha \\beta} = \\mathbf{g}_\\alpha \\cdot \\mathbf{g}_\\beta$. The deformation gradient \\citep{ogden2001} may be given by \\citep{haughton2001},\n\\begin{equation}\n\\mathbf{F} = \\pafrac{\\mathbf{x}}{\\mathbf{X}}=\\pafrac{\\mathbf{y}}{\\mathbf{Y}} + \\zeta^3_{,3}(\\xi^1,\\xi^2,0) \\mathbf{n} \\otimes \\mathbf{N},\n\\end{equation}\nfrom which we conclude that $\\zeta^3_{,3}(\\xi^1,\\xi^2,0)$ is a principal stretch. We may write the two-dimensional deformation gradient as,\n\\begin{equation}\\label{defgradx}\n\\pafrac{\\mathbf{y}}{\\mathbf{Y}} = \\mathbf{g}_\\alpha \\otimes \\mathbf{G}^{\\alpha},\n\\end{equation}\nwhich can then be directly computed. Following \\citet{haughton2001} we now define an orthonormal set of basis vectors $(\\mathbf{a_i})$ by,\n\\begin{equation}\n\\mathbf{a}_\\alpha = \\frac{\\mathbf{g}_\\alpha}{\\left| \\mathbf{g}_\\alpha \\right|} \\;\\;\\;\\; \\textrm{no sum},\n\\end{equation}\nwhich enables us to dispense with the distinction between covariant and contravariant tensors when we use $\\mathbf{a}_\\alpha$ as a basis \\citep{fu2001}.\nReferred to this basis, the general non-linear elasticity equilibrium equation $\\textrm{div} \\, \\hat{\\mbox{\\boldmath $\\sigma$}} = \\mathbf{0}$ becomes at zeroth order in $\\epsilon$, \\citep{haughton1978a, haughton2001}, \n\\begin{subequations}\\label{equilx}\n\\begin{gather}\n\\hat{\\sigma}_{\\mu \\kappa, \\mu} + \\hat{\\sigma}_{\\mu \\kappa} \\mathbf{a}_{\\mu,\\nu} + \\hat{\\sigma}_{\\mu \\nu} \\mathbf{a}_\\kappa \\cdot \\mathbf{a}_{\\mu, \\nu} + \\hat{h}_{, \\mu} \\hat{\\sigma}_{\\mu \\kappa}\/\\hat{h} = 0, \\quad \\kappa=1,2 \\label{equilx1}\\\\\n\\hat{\\sigma}_{\\mu \\kappa} \\mathbf{a}_3 \\cdot \\mathbf{a}_{\\mu, \\nu} + \\hat{P}\/\\hat{h} =0\\label{equilx2}\n\\end{gather}\n\\end{subequations}\nwhere $\\hat{\\sigma}_{\\alpha \\beta}$ is the $(\\alpha,\\beta)$-th element of the stress tensor $\\mbox{\\boldmath $\\sigma$}$, $\\hat{P}$ is a hydrostatic pressure applied to the inner surface and due to the use of the orthonormal basis vectors we have the additional definition $()_{,\\mu} = \\left| \\pafrac{\\mathbf{y}}{\\zeta^\\mu} \\right|^{-1} \\pafrac{()}{\\xi^\\mu}$. The third principal stress, $\\hat{\\sigma}_{33}$, is of order $\\epsilon$ at most, and therefore does not appear in \\rr{equibeqn}. This is often called the membrane assumption, $\\hat{\\sigma}_{33} = 0$.\n\nIt was shown by \\citet{naghdi1977} that the equilibrium equations presented above may be shown to be equivalent to those derived by considering the two-dimensional surface theory, under certain assumptions on $\\hat{P}$ and $\\hat{\\sigma}_{ij}$. This may be verified by a direct calculation of the equilibrium equations using the membrane theory detailed by \\citet{steigmann1999} or \\citet{steigmann2001} for instance. See \\citet{haughton2001} for more details on the comparison between the two theories.\n\n\nWe now use the coordinate systems introduced in \\rr{undefcon} and \\rr{defcon} to describe the undeformed and deformed configurations respectively. We may therefore associate the two curvilinear coordinate systems $(\\xi^\\alpha),(\\zeta^\\alpha)$ with the arclength and polar angle,\n\\begin{equation}\n\\xi^1 = A \\Theta, \\xi^2 = S, \\;\\; \\zeta^1 = A \\theta, \\zeta^2 = S,\n\\end{equation}\nwhere $A$ is a representative length. The orthonormal basis then becomes,\n\\begin{equation}\\label{adef}\n\\mathbf{a}_1 = \\mathbf{e}_\\theta, \\; \\mathbf{a}_2 = \\frac{\\hat{r}' \\mathbf{e}_r + \\hat{z}' \\mathbf{e}_z}{(\\hat{r}'^2+\\hat{z}'^2)^{1\/2}}, \\; \\mathbf{a}_3 = \\mathbf{a}_1 \\wedge \\mathbf{a}_2  = \\frac{\\hat{z}' \\mathbf{e}_r - \\hat{r}' \\mathbf{e}_z}{(\\hat{r}'^2+\\hat{z}'^2)^{1\/2}},\n\\end{equation}\nwhere a prime represents differentiation with respect to $\\hat{S}$. It may be seen that these vectors are unit vectors in the meridional, tangential and normal directions of the deformed surface. A similar orthonormal basis $\\mathbf{A}_\\alpha$ may be described with respect for the undeformed configuration with upper case letters replacing all equivalent lower case ones. The deformation gradient \\rr{defgradx} is then given by,\n\\begin{align}\n\\mathbf{F} &= \\frac{\\hat{r}}{\\hat{R}} \\mathbf{e}_\\theta \\otimes \\mathbf{e}_\\theta + \\left(\\hat{r}' \\mathbf{e}_r + \\hat{z}' \\mathbf{e}_z \\right) \\otimes \\left(\\frac{\\hat{R}' \\mathbf{e}_r + \\hat{Z}' \\mathbf{e}_z}{\\hat{R}'^2 + \\hat{Z}'^2}\\right) + \\frac{\\hat{h}}{\\hat{H}} \\mathbf{n} \\otimes \\mathbf{N} \\nonumber \\\\\n&=\\frac{\\hat{r}}{\\hat{R}} \\mathbf{e}_\\theta \\otimes \\mathbf{e}_\\theta+\n\\frac{\\sqrt{\\hat{r}'^2+\\hat{z}'^2}}{\\sqrt{\\hat{R}'^2+\\hat{Z}'^2}} \\left(\\frac{\\hat{r}' \\mathbf{e}_r + \\hat{z}' \\mathbf{e}_z}{\\sqrt{\\hat{r}'^2+\\hat{z}'^2}}\\right) \\otimes \\left(\\frac{\\hat{R}' \\mathbf{e}_r + \\hat{Z}' \\mathbf{e}_z}{\\sqrt{\\hat{R}'^2+\\hat{Z}'^2}}\\right) + \\frac{\\hat{h}}{\\hat{H}} \\mathbf{n} \\otimes \\mathbf{N}.\\label{defff}\n\\end{align}\nwhich shows that the principal directions are the meridional, tangential and normal directions respectively. As the deformation gradient can be written in a diagonal form, the stress tensor must also be diagonal with respect to the same basis, and so the only non-zero terms of $\\hat{\\mbox{\\boldmath $\\sigma$}}$ relative to this basis are $\\hat{\\sigma}_{11}$ and $\\hat{\\sigma}_{22}$. Additionally, using this basis we may immediately see the principal stretches in the three directions as the coefficients in \\rr{defff}.\nUsing \\rr{adef} in \\rr{equilx}, along with some rearrangement and noting that $()_{,2}$ in \\rr{equilx} becomes $()'\/\\sqrt{\\hat{r}'^2 + \\hat{z}'^2}$, we find the equilibrium equations, \n\\begin{gather}\\label{equibeqn}\n\\pafrac{(\\hat{h} \\hat{r} \\hat{\\sigma}_{11})}{\\theta} =0, \\\\\n\\pafrac{(\\hat{h} \\hat{r} \\hat{\\sigma}_{22})}{S} =  \\hat{h} \\hat{r}' \\hat{\\sigma}_{11},\\\\\n\\hat{\\sigma}_{11} \\frac{\\hat{z}'}{r \\sqrt{\\hat{r}'^2+\\hat{z}'^2}} + \\hat{\\sigma}_{22} \\frac{\\hat{z}'' \\hat{r}' - \\hat{r}'' \\hat{z}'}{(\\hat{r}'^2+\\hat{z}'^2)^{3\/2}} - \\frac{\\hat{P}}{\\hat{h}} = 0.\n\\end{gather}\nThe first equation in \\rr{equibeqn} is immediately satisfied due to the restriction of axisymmetry in the deformed configuration. The remaining two equations are the governing equations for the deformation, and represent force balances in the tangential and normal directions respectively. In \\rr{equibeqn}$_3$ the coefficients of the stresses are actually the principal curvatures in the deformed configuration, \n\\begin{equation}\\label{curvatures}\n\\hat{\\kappa}_s =  \\frac{1}{\\hat{r}'}\\left(\\frac{\\hat{z}'}{\\sqrt{\\hat{r}'^2+\\hat{z}'^2}}\\right)' = \\frac{-1}{\\hat{z}'}\\left(\\frac{\\hat{r}'}{\\sqrt{\\hat{r}'^2+\\hat{z}'^2}}\\right)' , \\; \\hat{\\kappa}_\\theta = \\frac{\\hat{z}'}{\\hat{r} \\sqrt{\\hat{r}'^2+\\hat{z}'^2}},\n\\end{equation}\nwhere $\\kappa_s, \\kappa_\\theta$ are the principal curvatures in the circumferential and radial directions respectively. These curvatures may be found from the components of the embedded deformed curvature tensor $\\mbox{\\boldmath $\\kappa$} = \\kappa_{\\alpha \\beta} \\mathbf{G}^{\\alpha} \\otimes \\mathbf{G}^{\\beta}$ using the definition $\\kappa_{\\alpha \\beta} = \\mathbf{n} \\cdot \\mathbf{g}_{\\alpha, \\beta}$, and then referred to the appropriate basis by $\\kappa_\\theta = \\kappa_{11} g^{11}, \\kappa_s = \\kappa_{22} g^{22}$.\n\nWhen the undeformed surface is flat or spherical we have $\\sqrt{\\hat{R}'^2 + \\hat{Z}'^2} = 1$ and the radicals in equation \\rr{curvatures} may be expressed neatly including the principal stretch $\\lambda_s$, but for a general surface this is not the case. We also note that the curvatures are always related by Codazzi's equation,\n\\begin{equation}\\label{codazzi2}\n(\\hat{r} \\hat{\\kappa}_\\theta)' = \\hat{r}' \\hat{\\kappa}_s. \n\\end{equation}\nwhich enables us to integrate one of the equilibrium equations under certain conditions, see \\rr{intequib}.\n\n\\clearpage\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}