Journal of the
Korean Mathematical Society JKMS

Let $M_c=M(2,0,c)$ be the moduli space of $\mathcal O(1)$-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0$ and $c_2=c$ on a K3 surface $X$, where $\mathcal O(1)$ is a generic ample line bundle on $X$. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a holomorphic symplectic structure on the smooth locus. In particular, $M_c$ has trivial canonical divisor. In [22], O'Grady asks if there is any symplectic desingularization of $M_{2n}$ for $n\ge 3$. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq 3$. This obviously implies that there is no symplectic desingularization.

Keywords: crepant resolution, irreducible symplectic variety, moduli space, sheaf, K3 surface, desingularization, Hodge-Deligne polynomial, Poincar\'{e} polynomial, stringy E-function

MSC numbers: 14D20, 14J60, 53D30