Arithmetic of singular character varieties and their $E$-polynomials

Abstract

We calculate the $E$-polynomials of the $S{L}_3(\mathbb{C})$ and $G{L}_3(\mathbb{C})$-character varieties of compact oriented surfaces of any genus and the $E$-polynomials of the $S{L}_2(\mathbb{C})$ and $G{L}_2(\mathbb{C})$-character varieties of compact non-orientable surfaces of any Euler characteristic. Our methods also give a new and significantly simpler computation of the $E$-polynomials of the $S{L}_2(\mathbb{C})$-character varieties of compact orientable surfaces, which were computed by Logares, Muñoz and Newstead for genus $g=1,2$ and by Martinez and Muñoz for $g\ge 3$. Our technique is based on the arithmetic of character varieties over finite fields. More specifically, we show how to extend the approach of Hausel and Rodriguez-Villegas used for non-singular (twisted) character varieties to the singular (untwisted) case.

41 pages

References in corpus (1)

• On the E-polynomials of a family of Character Varieties

Cited by in corpus (13)

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• A lax monoidal Topological Quantum Field Theory for representation varieties
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• Intersection cohomology of the moduli space of Higgs bundles on a genus 2 curve
• Representation variety for the rank one affine group
• Intersection cohomology of rank two character varieties of surface groups
• Generating series for the $E$-polynomials of $GL(n,\mathbb{C})$-character varieties
• Pseudo-quotients of algebraic actions and their application to character varieties
• $e$-polynomials of character varieties
• The point counting problem in representation varieties of torus knots
• The Number of Monodromy Representations of Abelian Varieties of Low $p$-Rank