Poisson cohomology of holomorphic toric Poisson manifolds. I

Abstract

A holomorphic toric Poisson manifold is a smooth toric variety equipped with a holomorphic Poisson structure, which is invariant under the torus action. In this paper, we compute the Poisson cohomology groups for all holomorphic toric Poisson structures on CPn, with the stand Poisson structure on CPn as a special case. We also compute the algebraic and the formal Poisson cohomology groups of holomorphic toric Poisson structures on Cn.