Poisson resolutions

Abstract

A resolution Z → X of a Poisson variety X is called 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 smooth Poisson surface S, the natural resolution S [n ] → S (n ) is a Poisson resolution. Furthermore, if ∣−KS ∣ is base-point-free, we prove that this is the unique projective Poisson resolution for S (n ).