Integrating Poisson manifolds via stacks

Details Integrating Poisson manifolds via stacks Integrating Poisson manifolds via stacks

2004-11-17

arxiv.org/abs/math/0411370v2

Travaux mathematiques, Volume 16 (2005), 285--297

Mathematics

Differential Geometry

Scientific paper

A symplectic groupoid $G.:=(G_1 \rightrightarrows G_0)$ determines a Poisson structure on $G_0$. In this case, we call $G.$ a symplectic groupoid of the Poisson manifold $G_0$. However, not every Poisson manifold $M$ has such a symplectic groupoid. This keeps us away from some desirable goals: for example, establishing Morita equivalence in the category of all Poisson manifolds. In this paper, we construct symplectic Weinstein groupoids which provide a solution to the above problem (Theorem \ref{main}). More precisely, we show that a symplectic Weinstein groupoid induces a Poisson structure on its base manifold, and that to every Poisson manifold there is an associated symplectic Weinstein groupoid.

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