Hodge-Deligne polynomials of SL(2,C)-character varieties for curves of small genus

TL;DR

This paper calculates the Hodge-Deligne polynomials of SL(2,C)-character varieties for small genus surfaces, introducing a new geometric stratification technique to analyze these moduli spaces.

Contribution

It presents a novel geometric method for computing Hodge-Deligne polynomials of representation spaces for small genus, considering various holonomy types.

Findings

Explicit formulas for Hodge-Deligne polynomials of the moduli spaces.

New stratification approach for analyzing representation spaces.

Insights into the structure of character varieties for small genus.

Abstract

We compute the Hodge-Deligne polynomials of the moduli spaces of representations of the fundamental group of a complex surface into SL(2,C), for the case of small genus g, and allowing the holonomy around a fixed point to be any matrix of SL(2,C), that is Id, -Id, diagonalisable, or of either of the two Jordan types. For this, we introduce a new geometric technique, based on stratifying the space of representations, and on the analysis of the behaviour of the Hodge-Deligne polynomial under fibrations.