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

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

Moduli spaces of bundles over non-projective K3 surfaces

We study moduli spaces of sheaves over non-projective K3 surfaces. More precisely, if $v=(r,\xi,a)$ is a Mukai vector on a K3 surface $S$ with $r$ prime to $\xi$ and $\omega$ is a "generic" K\"ahler class on $S$, we show that the moduli space $M$ of $\mu_{\omega}-$stable sheaves on $S$ with associated Mukai vector $v$ is an irreducible holomorphic symplectic manifold which is deformation equivalent to a Hilbert scheme of points on a K3 surface. If $M$ parametrizes only locally free sheaves, it is moreover hyperk\"ahler. Finally, we show that there is an isometry between $v^{\perp}$ and $H^{2}(M,\mathbb{Z})$ and that $M$ is projective if and only if $S$ is projective.Comment: 42 pages; major revisions; to appear in Kyoto J. Mat

The 2-Factoriality of the O'Grady Moduli Spaces

The aim of this work is to show that the moduli space $M_{10}$ introduced by O'Grady in \cite{OG1} is a $2-$factorial variety. Namely, $M_{10}$ is the moduli space of semistable sheaves with Mukai vector $v:=(2,0,-2)\in H^{ev}(X,\mathbb{Z})$ on a projective K3 surface $X$. As a corollary to our construction, we show that the Donaldson morphism gives a Hodge isometry between $v^{\perp}$ (sublattice of the Mukai lattice of $X$) and its image in $H^{2} (\widetilde{M}_{10},\mathbb{Z})$, lattice with respect to the Beauville form of the $10-$dimensional irreducible symplectic manifold $\widetilde{M}_{10}$, obtained as symplectic resolution of $M_{10}$. Similar results are shown for the moduli space $M_{6}$ introduced by O'Grady in \cite{OG2}.Comment: 26 page

Kahlerness of moduli spaces of stable sheaves over non-projective K3 surfaces

We show that a moduli space of slope-stable coherent sheaves over a K3 surface is a compact hyperkahler manifold if and only if its second Betti number is the sum of its Hodge numbers h^{2,0}, h^{1,1} and h^{0,2}.