E-polynomial of SL(2,C)-character varieties of complex curves of genus 3

TL;DR

This paper calculates the E-polynomials of moduli spaces of SL(2,C) representations of genus 3 complex curves, extending previous results for lower genus using stratification and fibration analysis techniques.

Contribution

It provides the first computation of E-polynomials for genus 3, advancing the understanding of character varieties for higher genus curves.

Findings

E-polynomials computed for genus 3 character varieties

Extended methods from genus 1,2 to genus 3

Insights into the structure of moduli spaces

Abstract

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve of genus g=3 into $ SL(2,C), and also of the moduli space of twisted representations. The case of genus g=1,2 has already been done in [http://arxiv.org/abs/1106.6011]. We follow the geometric technique introduced in [http://arxiv.org/abs/1106.6011], based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.