Poisson deformations of symplectic quotient singularities

Abstract

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 G⊂ Sp( V) 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 •(X, C) of any smooth symplectic resolution X↠V/G (multiplicative McKay correspondence). We prove further that if G⊂GL( h) is an irreducible Weyl group and V= h⊕ h ∗ , then no smooth symplectic resolution of V/ G exists unless G is of types A , B , C .