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 ${\tilde{M}}_v$. A particular case is the $10-$dimensional O’Grady example ${\tilde{M}}_{10}$ of irreducible symplectic manifold. We show that ${\tilde{M}}_v$ is an irreducible symplectic manifold which is deformation equivalent to ${\tilde{M}}_{10}$ and that $H^2(M_v,\mathbb{Z})$ is Hodge isometric to the sublattice $v^{\perp }$ of the Mukai lattice of $S$. Similar results are shown when $S$ is an abelian surface.