A global Weinstein splitting theorem for holomorphic Poisson manifolds.

Abstract: A foundational result in Poisson geometry, due to Weinstein, states that any Poisson bracket on a manifold can be written locally as the Poisson bracket of symplectic form in canonical coordinates, and a Poisson bracket that vanishes at a point.  A key consequence is that every Poisson manifold has a canonical foliation with symplectic leaves. I will give an introduction to these ideas, and then discuss the problem of globalizing Weinstein's decomposition, to split the manifold itself (or a covering thereof) as a product of a symplectic leaf and a transverse Poisson manifold.  While the existence of such a splitting is rare in the context of smooth manifolds, it turns out to be automatic for holomorphic Poisson structures on compact Kähler manifold admitting a simply-connected compact symplectic leaf.  This talk is based on joint work with Stéphane Druel, Jorge Vitório Pereira, and Frédéric Touzet, which in turn relies in an essential way on a notion of "subcalibrations" in Poisson geometry introduced recently by Pedro Frejlich and Ioan Mărcuț.

MathematicsPhysics

Audience: researchers in the discipline

Seminario de Geometría y Física - Matemática UCN-USP