Mixed Hodge polynomials of character varieties

Details Mixed Hodge polynomials of character varieties Mixed Hodge polynomials of character varieties

2006-12-21

arxiv.org/abs/math/0612668v3

Mathematics

Algebraic Geometry

with an appendix by Nicholas M. Katz; 57 pages. revised version: New definition for homogeneous weight in Definition 4.1.6, su

Scientific paper

We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) 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 M_n: 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.

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