Kiem proved that the moduli space of SL(2,C)-Higgs bundles over a smooth projective curve X admits a symplectic desingularization if and only if g(X) = 2. Based on work of Schedler and Bellamy, Tirelli reproduced this result to the case of GL(2, C)-Higgs bundles, proving as well that no symplectic desingularization of SL(n, C) and GL(n, C)-Higgs bundles exists besides these two cases where (n, g) = (2, 2).
Using the normal cone degeneration, Donagi, Ein and Lazarsfeld provided a degeneration of the moduli space of sheaves on a K3 surface into the moduli space of GL(n,C)-Higgs bundles of a curve X embedded in the K3. We generalize this construction to the case of a hyperelliptic curve inside its Jacobian, obtaining a degeneration of the moduli space of sheaves with fixed determinant and dual determinant to the moduli space of SL(n, C)-Higgs bundles.
(On genus 2) we use these degenerations to obtain a degeneration of the symplectically desingu- larized SL(2,C) and GL(2,C)-Higgs moduli spaces into, respectively, O’Grady’s 6-dimensional and 10-dimensional examples of irreducible holomorphic symplectic manifolds.