Nonexistence of a crepant resolution of some moduli spaces of sheaves on a K3 surface

TL;DR

This paper proves that certain moduli spaces of sheaves on K3 surfaces do not admit crepant resolutions, indicating the impossibility of symplectic desingularizations for these singular varieties.

Contribution

It demonstrates the nonexistence of crepant resolutions for specific moduli spaces of sheaves on K3 surfaces when the second Chern class is sufficiently large.

Findings

No crepant resolution exists for $M_{2n}$ when $n extgreater=3$

Impossibility of symplectic desingularization for these moduli spaces

Singular moduli spaces lack smooth symplectic resolutions

Abstract

Let $M_c=M(2,0,c)$ be the moduli space of O(1)-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0$ and $c_2=c$ on a K3 surface $X$ where O(1) is a generic ample line bundle on $X$. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a symplectic structure on the smooth locus. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq 3$. This implies that there is no symplectic desingularization.