A global Weinstein splitting theorem for holomorphic Poisson manifolds

TL;DR

This paper proves a global splitting theorem for holomorphic Poisson manifolds with certain symplectic leaves, linking their structure to universal covers and finite covers, and also addresses a special case of Beauville's conjecture.

Contribution

It establishes a global Weinstein splitting theorem for holomorphic Poisson manifolds under specific conditions, advancing the understanding of their geometric structure.

Findings

Decomposition of compact Kähler Poisson manifolds after finite étale cover

Verification of a special case of Beauville's conjecture on tangent bundle splitting

Structural insights into Poisson manifolds with symplectic leaves having finite fundamental group

Abstract

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.