Deformation of the O'Grady moduli spaces

In this paper we study moduli spaces of sheaves on an abelian or projective K3 surface. If S is a K3, v = 2w is a Mukai vector on S, where w is primitive and w^{2}= 2, and H is a v-generic polarization on S, then the moduli space M_v of H-semistable sheaves on S whose Mukai vector is v admits a symplectic resolution \widetilde{M}_v. A particular case is the 10-dimensional O'Grady example \widetilde{M}_10 of an irreducible symplectic manifold. We show that \widetilde{M}_v is an irreducible symplectic manifold which is deformation equivalent to \widetilde{M}_10 and that H2(M_v,\mathbb{Z}) is Hodge isometric to the sublattice v^{\perp} of the Mukai lattice of S. Similar results are shown when S is an abelian surface. © Walter de Gruyter Berlin · Boston 2013.