Intersection cohomology of the moduli space of Higgs bundles on a genus 2 curve

TL;DR

This paper computes the intersection cohomology of the moduli space of Higgs bundles on a genus 2 curve, showing the mixed Hodge structure is pure, extending known results from smooth cases to singular moduli spaces.

Contribution

It constructs a semismall desingularization of the moduli space and uses the decomposition theorem to compute intersection cohomology, revealing purity of the mixed Hodge structure.

Findings

Intersection cohomology computed for the moduli space.

Mixed Hodge structure is shown to be pure.

Method extends results from smooth to singular moduli spaces.

Abstract

Let $C$ be a smooth projective curve of genus $2$. Following a method by O' Grady, we construct a semismall desingularization $\tilde{\mathcal{M}}_{Dol}^G$ of the moduli space $\mathcal{M}_{Dol}^G$ of semistable $G$-Higgs bundles of degree 0 for $G=GL(2,\mathbb{C}), SL(2,\mathbb{C})$. By the decomposition theorem by Beilinson, Bernstein, Deligne one can write the cohomology of $\tilde{\mathcal{M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $\mathcal{M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $\mathcal{M}_{Dol}^G$ and prove that the mixed Hodge structure on it is actually pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.