Poisson deformations of affine symplectic varieties

Abstract

We prove that the Poisson deformation functor of an affine (singular) symplectic variety is unobstructed. As a corollary, we prove the following result. For an affine symplectic variety $X$ with a good $C^{*}$-action (where its natural Poisson structure is positively weighted), the following are equivalent.

(1) $X$ has a crepant projective resolution.

(2) $X$ has a smoothing by a Poisson deformation.

A typical example is (the normalization) of a nilpotent orbit closure in a complex simple Lie algebra. By the theorem, one can see which orbit closure has a smoothing by a Poisson deformation.