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\subset 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 rational cohomology H^*(X) of any smooth symplectic resolution X \to V/G (multiplicative McKay correspondence). We prove further that if G is an irreducible Weyl group in GL(h) and V=h+ h^* then no smooth symplectic resolution of V/G exists unless G is of types A,B, or C.
Keywords
Cite
@article{arxiv.math/0212279,
title = {Poisson deformations of symplectic quotient singularities},
author = {Victor Ginzburg and Dmitry Kaledin},
journal= {arXiv preprint arXiv:math/0212279},
year = {2010}
}
Comments
A gap pointed out by G. Bellamy is fixed in a new Remark 5.9