A global Weinstein splitting theorem for holomorphic Poisson manifolds

Abstract

We prove that if a compact Kähler Poisson manifold has a compact symplectic leaf with finite fundamental group, then, after passing to a finite étale cover, it decomposes as the product of the universal cover of the leaf and some other Poisson manifold. As a step in the proof, we establish a special case of Beauville’s conjecture on the structure of compact Kähler manifolds with split tangent bundle.