Holomorphic symplectic manifolds from semistable Higgs bundles

TL;DR

This paper constructs new 18- and 12-dimensional holomorphic symplectic varieties from moduli spaces of semistable Higgs bundles on a genus three hyperelliptic curve, revealing their crepant resolutions and singularity models.

Contribution

It demonstrates that quotients of specific Higgs bundle moduli spaces by hyperelliptic involutions form new holomorphic symplectic varieties with crepant resolutions, connecting to known singularity models.

Findings

The 18-dimensional quotient admits a crepant resolution.

The 12-dimensional quotient also admits a crepant resolution.

The local models relate to O'Grady's singularities studied by Kaledin and Lehn.

Abstract

Let $\mathcal{M}_{C}(2, 0)$ be the moduli space of semistable rank two and degree zero Higgs bundles on a smooth complex hyperelliptic curve $C$ of genus three. We prove that the quotient of $\mathcal{M}_{C}(2, 0)$ by a twisted version of the hyperelliptic involution is an 18-dimensional holomorphic symplectic variety admitting a crepant resolution, whose local model was studied by Kaledin and Lehn to describe O'Grady's singularities. Similarly, by considering the moduli space of Higgs bundles with trivial determinant $\mathcal{M}_C(2, \mathcal{O}_{C})\subseteq \mathcal{M}_C(2, 0)$, we show that the quotient of $\mathcal{M}_C(2, \mathcal{O}_{C})$ by the hyperelliptic involution is a 12-dimensional holomorphic symplectic variety admitting a crepant resolution.