Singular symplectic moduli spaces
  Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces found by O'Grady. Consequently, since singular moduli space that do not belong to these exceptional cases have singularities in codimension $\geq4$ they do no admit projective symplectic resolutions. 
Dmitry Kaledin (IUM), Manfred Lehn (MI), Christoph Sorger (LMJL)