Intersection cohomology of rank two character varieties of surface groups

TL;DR

This paper computes intersection cohomology invariants of rank two character varieties and Higgs moduli spaces for surface groups, providing new insights into their geometric and topological structures.

Contribution

It explicitly calculates intersection E- and Poincaré polynomials for specific character varieties and Higgs moduli spaces, advancing understanding of their intersection cohomology.

Findings

Computed intersection E-polynomials for GL_2, SL_2, PGL_2 character varieties

Derived intersection Poincaré polynomials for these moduli spaces

Provided results relevant to the P=W conjectures for singular moduli spaces

Abstract

For $G = \mathrm{GL}_2, \mathrm{SL}_2, \mathrm{PGL}_2$ we compute the intersection E-polynomials and the intersection Poincar\'e polynomials of the $G$-character variety of a compact Riemann surface $C$ and of the moduli space of $G$-Higgs bundles on $C$ of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.