On Hodge polynomials of Singular Character Varieties

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.

To appear in the Proceedings of the Special Session on Complex Geometry of the ISAAC conference, Aveiro, Portugal (2019)