E-polynomials of $\mathrm{SL}(2, \mathbb{C})$-character varieties of complex curves of genus $3$

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 $\mathrm{SL}(2, \mathbb{C})$, and also of the moduli space of twisted representations. The case of genus $g=1, 2$ has already been done in [12]. We follow the geometric technique introduced in [12], based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.