Poisson deformations of affine symplectic varieties II

TL;DR

This paper proves that the map between Poisson deformation spaces of an affine symplectic variety and its crepant resolution is a Galois covering, extending previous results and constructing universal deformations of nilpotent orbit closures.

Contribution

It establishes that the Poisson deformation map is a Galois covering, generalizing Markman's compact case result and explicitly constructing universal deformations for certain nilpotent orbit closures.

Findings

The Poisson deformation map is a Galois covering.

Both deformation spaces are non-singular.

Explicit universal Poisson deformations are constructed for specific nilpotent orbit closures.

Abstract

This is a continuation of math.AG/0609741. Let Y be an affine symplectic variety with a C^*-action with positive weights, and let \pi: X -> Y be its crepant resolution. Then \pi induces a natural map PDef(X) -> PDef(Y) of Kuranishi spaces for the Poisson deformations of X and Y. In the Part I, we proved that PDef(X) and PDef(Y) are both non-singular, and this map is a finite surjective map. In this paper (Part II), we prove that it is a Galois covering. Markman already obtained a similar result in the compact case, which was a motivation of this paper. As an application, we shall construct explicitly the universal Poisson deformation of the normalization \tilde{O} of a nilpotent orbit closure \bar{O} in a complex simple Lie algebra when \tilde{O} has a crepant resolution.