Character Varieties
Character Varieties
复制标题
性格品种
DOI：
10.1090/s0002-9947-2012-05448-1
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
Adam S. Sikora
中科院分区：
文献类型：
--
作者：
Adam S. Sikora
在Hom（Gamma，G）上G-作用的共轭几何不变理论的背景下，研究了Hom（Gamma，G）上的G-作用的不可约和完全可约表示的性质.特别地，我们研究了特征标簇X_G（Gamma）=Hom（Gamma，G）//G的性质.本文用Gamma的一阶上同调群来刻画X_G（Gamma）的切空间，推广了著名的公式.设M是可定向的3-流形，其连通边界F的亏格>1，设X_G^g（F）是F的G -特征簇的子集，该簇由良好表示的共轭类组成。根据Goldman定理，X_G^g（F）是一个全纯辛流形.我们证明了pi_1（F）的好G-表示集扩张到pi_1（M）的表示是X_G^g（F）的迷向子流形。如果这些表示对应于M的G-特征簇的约化点，则这个子流形是拉格朗日的。
We study properties of irreducible and completely reducible representations of finitely generated groups Gamma into reductive algebraic groups G in in the context of the geometric invariant theory of the G-action on Hom(Gamma,G) by conjugation. In particular, we study properties of character varieties, X_G(Gamma)=Hom(Gamma,G)//G. We describe the tangent spaces to X_G(Gamma) in terms of first cohomology groups of Gamma with twisted coefficients, generalizing the well known formula. Let M be an orientable 3-manifold with a connected boundary F of genus>1 and let X_G^g(F) be the subset of the G -character variety of F composed of conjugacy classes of good representations. By a theorem of Goldman, X_G^g(F) is a holomorphic symplectic manifold. We prove that the set of good G-representations of pi_1(F) which extend to representations of pi_1(M) is an isotropic submanifold of X_G^g(F). If these representations correspond to reduced points of the G-character variety of M then this submanifold is Lagrangian.