Mixed Hodge polynomials of character varieties With an appendix by Nicholas M .
نویسندگان
چکیده
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.
منابع مشابه
Mixed Hodge polynomials of character varieties
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 PG...
متن کاملMirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve
This is a survey of results and conjectures on mirror symmetry phenomena in the nonAbelian Hodge theory of a curve. We start with the conjecture of Hausel–Thaddeus which claims that certain Hodge numbers of moduli spaces of flat SL(n, C) and PGL(n, C)connections on a smooth projective algebraic curve agree. We then change our point of view in the non-Abelian Hodge theory of the curve, and conce...
متن کاملArithmetic Harmonic Analysis on Character and Quiver Varieties
We propose a general conjecture for the mixed Hodge polynomial of the generic character varieties of representations of the fundamental group of a Riemann surface of genus g to GLn.C/ with fixed generic semisimple conjugacy classes at k punctures. This conjecture generalizes the Cauchy identity for Macdonald polynomials and is a common generalization of two formulas that we prove in this paper....
متن کاملm at h . A G ] 1 3 N ov 2 00 4 Mixed Hodge structure of global Brieskorn
In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial f in C, where f satisfies a certain regularity condition at infinity (and hence has isolated singularities). We give an algorithm which produces a basis of a localization of the Brieskorn module which is compatible with its mixed Hodge structure. As an application we show that the notion of a Hodge cy...
متن کاملJ ul 2 00 4 Mixed Hodge structure of global Brieskorn modules
In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial f in C, where f satisfies a certain regularity condition at infinity (and hence has isolated singularities). We give an algorithm which produces a basis of a localization of the Brieskorn module which is compatible with its mixed Hodge structure. As an application we show that the notion of a Hodge cy...
متن کامل