Deformation of the O'Grady moduli spaces

TL;DR

This paper investigates the deformation and lattice structure of certain moduli spaces of sheaves on K3 and abelian surfaces, revealing their symplectic resolutions are deformation equivalent to O'Grady's example.

Contribution

It demonstrates that the symplectic resolutions of these moduli spaces are deformation equivalent to O'Grady's 10-dimensional example and describes their second cohomology lattice structure.

Findings

Symplectic resolutions are deformation equivalent to O'Grady's example.

The second cohomology is Hodge isometric to a sublattice of the Mukai lattice.

Results hold for both K3 and abelian surfaces.

Abstract

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.