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.
Categories: math.AG