Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties

TL;DR

This paper investigates the singularities of moduli spaces of sheaves on K3 surfaces using Nakajima quiver varieties, revealing local isomorphisms and symplectic resolutions related to stability conditions.

Contribution

It establishes a local isomorphism between moduli spaces of pure dimension one sheaves on K3 surfaces and quiver varieties, linking symplectic resolutions to GIT variations.

Findings

Moduli spaces are locally isomorphic to quiver varieties.

Symplectic resolutions correspond to GIT quotient variations.

Singularities depend on polarization choices.

Abstract

The aim of this paper is to study the singularities of certain moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non--generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. For sheaves that are pure of dimension one, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety and that, via this isomorphism, the natural symplectic resolutions correspond to variations of GIT quotients of the quiver variety.