Poisson resolutions

Abstract

A resolution $Z\to X$ of a Poisson variety $X$ is called {\em Poisson} if every Poisson structure on $X$ lifts to a Poisson structure on $Z$. For symplectic varieties, we prove that Poisson resolutions coincide with symplectic resolutions. It is shown that for a Poisson surface $S$, the natural resolution $S^{[n]}\to S^{(n)}$ is a Poisson resolution. Furthermore, if $Bs|-K_S|=\varnothing$, we prove that this is the unique projective Poisson resolution for $S^{(n)}$.

some changes in section 5