A global Weinstein splitting theorem for holomorphic Poisson manifolds

Resumen:

Weinstein proved that any point in a Poisson manifold has a neighbourhood that decomposes as a product of a symplectic manifold and a Poisson manifold for which the Poisson barcket vanishes at the marked point. As a consequence, any Poisson manifold carries a canonical foliation by even-dimensional submanifolds, called its symplectic leaves. In joint work with B. Pym, J. V. Pereira and F. Touzet, 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.

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