On moduli spaces of sheaves on K3 or abelian surfaces

TL;DR

This paper studies the structure of moduli spaces of sheaves on K3 and abelian surfaces, revealing constraints on the existence of certain symplectic manifolds and clarifying their geometric properties.

Contribution

It provides new results on the non-existence of certain irreducible symplectic manifolds related to moduli spaces of sheaves on K3 surfaces.

Findings

Excludes the existence of new projective irreducible symplectic manifolds birationally related to these moduli spaces.

Clarifies the structure of moduli spaces of sheaves on K3 and abelian surfaces.

Provides conditions under which symplectic resolvability is achieved.

Abstract

We investigate the moduli spaces of one- and two-dimensional sheaves on projective K3 and abelian surfaces that are semistable with respect to a nongeneral ample divisor with regard to the symplectic resolvability. We can exclude the existence of new examples of projective irreducible symplectic manifolds lying birationally over components of the moduli spaces of one-dimensional semistable sheaves on K3 surfaces, and over components of many of the moduli spaces of two-dimensional sheaves on K3 surfaces, in particular, of those for rank two sheaves.