The moduli spaces of sheaves on K3 surfaces are irreducible symplectic varieties

We show that the moduli spaces of sheaves on a projective K3 surface are irreducible symplectic varieties, and that the same holds for the fibers of the Albanese map of moduli spaces of sheaves on an Abelian surface.Comment: 59 page

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 irreducible symplectic manifold. We show that $\widetilde{M}_{v}$ is an irreducible symplectic manifold which is deformation equivalent to $\widetilde{M}_{10}$ 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.Comment: 29 page

On factoriality of threefolds with isolated singularities

We investigate the existence of complete intersection threefolds $X \subset \mathbb{P}^n$ with only isolated, ordinary multiple points and we provide some sufficient conditions for their factoriality.Comment: 18 pages. To appear in the Michigan Mathematical Journa

The Hodge numbers of O'Grady 10 via Ng\^o strings

We determine the Hodge numbers of the hyper-K\"ahler manifold known as O'Grady 10 by studying some related modular Lagrangian fibrations by means of a refinement of the Ng\^o Support Theorem.Comment: Revised and final version to appear in Jour. Math. Pur. et App