On the types of the mixed Hodge structures of character varieties

TL;DR

This paper proves that the mixed Hodge structures of character varieties are of Hodge--Tate type and are independent of eigenvalue choices, also showing the purity of structures in related moduli spaces.

Contribution

It confirms a conjecture about the Hodge--Tate nature and independence of mixed Hodge polynomials, and establishes purity for certain moduli spaces.

Findings

Mixed Hodge structures of character varieties are Hodge--Tate.

Mixed Hodge polynomials are independent of generic eigenvalues.

Moduli spaces of semistable parabolic Higgs bundles are pure.

Abstract

In this paper, we show that the mixed Hodge structures of character varieties are of Hodge--Tate type and that the mixed Hodge polynomials are independent of the choice of generic eigenvalues, which is a conjecture due to Hausel, Letellier and Rodriguez-Villegas. Moreover, we investigate the mixed Hodge structures of the moduli space of semistable parabolic Higgs bundles and the moduli space of semistable regular singular parabolic connections. We show that the mixed Hodge structures of these moduli spaces are pure.