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摘要翻译：
本文研究实辛群Sp(2n,R）中曲面群（即闭定向曲面的基本群）表示的模空间。模空间由一个整数不变量划分，称为托莱多不变量。这个不变量以Milnor-Wood型不等式为界。我们的主要结果是极大表示（即具有极大Toledo不变量的表示）的模空间的连通分量数的计数。我们的方法利用了文献ARXIV:0909.4487[Math.DG]中证明的非阿贝尔Hodge理论的对应性，将表示空间与多稳定Sp(2n，R)-Higgs束的模空间进行了识别。最大表示的新离散不变量的发现提供了一个关键步骤。这些新的不变量是由群GL(n，R)的Sp(2n，R)-Higgs丛的模空间与扭Higgs丛的模空间在极大情况下的一个辨识而来的。
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英文标题：
《Higgs bundles and surface group representations in the real symplectic
group》
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作者：
Oscar Garcia-Prada (CSIC, Madrid), Peter B. Gothen (Universidade do
Porto) and Ignasi Mundet i Riera (Universitat de Barcelona)
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最新提交年份：
2012
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Differential Geometry 微分几何
分类描述：Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis
复形，接触，黎曼，伪黎曼和Finsler几何，相对论，规范理论，整体分析
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英文摘要：
In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality. Our main result is a count of the number of connected components of the moduli space of maximal representations, i.e. representations with maximal Toledo invariant. Our approach uses the non-abelian Hodge theory correspondence proved in a companion paper arXiv:0909.4487 [math.DG] to identify the space of representations with the moduli space of polystable Sp(2n,R)-Higgs bundles. A key step is provided by the discovery of new discrete invariants of maximal representations. These new invariants arise from an identification, in the maximal case, of the moduli space of Sp(2n,R)-Higgs bundles with a moduli space of twisted Higgs bundles for the group GL(n,R).
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PDF链接：
https://arxiv.org/pdf/0809.0576
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