Flops and Poisson deformations of symplectic varieties

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 $X^{\mathrm{\prime }}$. If $X$ is non-singular, then $X^{\mathrm{\prime }}$ is non-singular, too. Moreover, when $Y$ has a good $C^{\ast }$-action, $X$ and $X^{\mathrm{\prime }}$ have the same kind of singularities.