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

Let M, be the moduli space of semistable torsion-free sheaves of rank 2 with Chern classes c(1) = 0 and c(2) = c over a K3 surface with generic polarization. When c = 2n >= 4 is even, M, 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.