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

Let J be an abelian surface with a generic ample line bundle . For n≥1, the moduli space M_J(2,0,2n) of (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 M_J(2,0,2n) for n≥2. This certainly implies that there is no symplectic desingularization of M_J(2,0,2n) for n≥2. Jaeyoo Choy was partially supported by KRF 2003-070-C00001 Young-Hoon Kiem was partially supported by a KOSEF grant R01-2003-000-11634-0.