A global Weinstein splitting theorem for holomorphic Poisson manifolds

We prove that if a compact K¨ahler Poisson manifold has a symplectic leaf with finite fundamental group, then after passing to a finite ´etale 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 K¨ahler manifolds with split tangent bundle.