Flops and Poisson deformations of symplectic varieties

TL;DR

This paper studies Poisson deformations of convex symplectic varieties, revealing that non-singular crepant terminalizations share similar singularity properties, especially under good $C^*$-actions, extending deformation theory beyond projective cases.

Contribution

It introduces a Poisson deformation approach for convex symplectic varieties, generalizing deformation theory to non-projective cases and establishing singularity preservation under certain conditions.

Findings

Non-singular crepant terminalizations are preserved under Poisson deformations.

Convex symplectic varieties can be effectively studied via Poisson schemes.

Singularity types are maintained when a good $C^*$-action exists.

Abstract

This is a local version of math.AG/0506534. We shall deal with the deformation of a convex symplectic variety $X$ instead of a projective one. The usual deformation does not work well in the convex case. Instead, we regard $X$ as a Poisson scheme and study its Poisson deformation. One of the application is the following: Let $Y$ be an affine symplectic variety, and assume that $Y$ has two $Q$-factorial crepant terminalizations $X$ and $X'$. If $X$ is non-singular, then $X'$ is non-singular, too. Moreover, when $Y$ has a good $C^*$-action, $X$ and $X'$ have the same kind of singularities.