Any Poisson bracket on a differentiable manifold can be written locally as the sum of two commuting Poisson brackets: a Poisson bracket of
a symplectic form and a Poisson bracket that vanishes at a point. This is the so-called Weinstein splitting. The result is strictly local; in
general, it is impossible to globalize the two local Poisson structures.  It turns out that for holomorphic Poisson structures on compact Kähler
manifolds admitting a simply connected compact symplectic leaf, the local Weinstein splitting globalizes and produces a global splitting of
a finite etale cover of the ambient manifold as a product of two compact Kähler Poisson manifolds.

Based on a joint work with Stéphane Druel, Brent Pym, and Frédéric Touzet.