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

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.

arXiv comment40 pages; final version; As pointed out to us by Z. Zhang, we prove quadraticity and not formality of the Kuranishi family. Quadraticity is all we need for our main theorem. The current version reflects this correction. A few other improvements in exposition and correction of typos