Mixed Hodge polynomials of character varieties

• 成果类型: Article
• 署名作者: Hausel, Tamas; Rodriguez-Villegas, Fernando
• 署名单位: University of Oxford
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-008-0142-x
• 发表日期: 2008
• 页码: 555-624
• 关键词: rank-2 higgs bundles; MODULI SPACES; cohomology ring; flat connections; invariant-theory; riemann surface; finite-fields; root systems; REPRESENTATIONS; Duality

摘要

We calculate the E-polynomials of certain twisted GL( n, C)character varieties Mn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie- type GL( n, Fq) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL( n, C)- character variety. The calculation also leads to several conjectures about the cohomology of Mn: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n = 2.