Holomorphic symplectic manifolds from semistable Higgs bundles

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.