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.