Holomorphic symplectic manifolds from semistable Higgs bundles

Let MC(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 MC(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 MC(2, OC)⊆ MC(2, 0), we show that the quotient of MC(2, OC) by the hyperelliptic involution is a 12-dimensional holomorphic symplectic variety admitting a crepant resolution.