On the existence of a symplectic desingularization of some moduli spaces of sheaves on a K3 surface

Abstract

Let M_c be the moduli space of semistable torsion-free sheaves of rank 2 with Chern classes c₁ = 0 and c₂ = c over a K3 surface with generic polarization. When $c=2n\ge 4$ is even, M_c is a singular projective variety which admits a symplectic form, called the Mukai form, on the smooth part. A natural question raised by O'Grady asks if there exists a desingularization on which the Mukai form extends everywhere nondegenerately. In this paper we show that such a desingularization does not exist for many even integers c by computing the stringy Euler numbers.