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 ∣−K_S ∣ is base-point-free, we prove that this is the unique projective Poisson resolution for S  ^{(n  )}.