Hodge structure on the cohomology of the moduli space of Higgs bundles

TL;DR

This paper computes the Hodge structure of the cohomology of the moduli space of Higgs bundles on a compact Riemann surface, demonstrating that it is pure of weight k, building on prior results about cohomology generation.

Contribution

It establishes that the natural Hodge structure on the cohomology of Higgs bundle moduli spaces is pure of weight k, extending previous cohomology generation results.

Findings

Hodge structure on cohomology is pure of weight k

Utilizes prior results to compute Hodge structures

Confirms the purity of the Hodge structure in this context

Abstract

We study the moduli space of Higgs bundles on a compact Riemann surface. It was shown by Thaddeus and Hausel (in rank 2) and Markman (in general rank) that the rational cohomology ring of this space is generated by universal classes. In this paper, we use their results to compute the Hodge structure on the cohomology of this moduli space. We show that the natural Hodge structure on H^k is pure of weight k.