On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface

Abstract

Let J be an abelian surface with a generic ample line bundle Open image in new window. For n≥1, the moduli space MJ(2,0,2n) of Open image in new window(1)-semistable sheaves F of rank 2 with Chern classes c1(F)=0, c2(F)=2n is a singular projective variety, endowed with a holomorphic symplectic structure on the smooth locus. In this paper, we show that there does not exist a crepant resolution of MJ(2,0,2n) for n≥2. This certainly implies that there is no symplectic desingularization of MJ(2,0,2n) for n≥2.