We consider the problem of quantization of smooth symplectic varieties in the algebro-geometric setting. We show that, under appropriate cohomological assumptions, the Fedosov quantization procedure goes through with minimal changes. The assumptions are satisfied, for example, for affine and for projective varieties. We also give a classification of all possible quantizations.

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

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety.In particular, let V be a finite-dimensional complex symplectic vector space and GCSpðV Þ a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V =G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the cohomology H ðX ; CÞ of any smooth symplectic resolution X 7V =G (multiplicative McKay correspondence). We prove further that if GCGLðhÞ is an irreducible Weyl group and V ¼ h"h à ; then no smooth symplectic resolution of V =G exists unless G is of types A; B; C: r

No abstract

No abstract