On Hodge polynomials of Singular Character Varieties

TL;DR

This paper develops new methods to compute Hodge-Deligne and Serre polynomials of complex reductive group character varieties, especially for free and free abelian groups, advancing understanding of their geometric invariants.

Contribution

It introduces novel techniques for calculating Hodge-Deligne and Serre polynomials of character varieties, with applications to free and free abelian groups.

Findings

New computational techniques for Hodge-Deligne polynomials

Applications to free and free abelian group cases

Framework for future detailed proofs

Abstract

Let $\mathcal{X}_{\Gamma}G:=\mathrm{Hom}(\Gamma,G)/\!/G$ be the $G$-character variety of $\Gamma$, where $G$ is a complex reductive group and $\Gamma$ a finitely presented group. We introduce new techniques for computing Hodge-Deligne and Serre polynomials of $\mathcal{X}_{\Gamma}G$, and present some applications, focusing on the cases when $\Gamma$ is a free or free abelian group. Detailed constructions and proofs of the main results will appear elsewhere.