Int\'egration symplectique des vari\'et\'es de Poisson r\'eguli\`eres

Abstract

A symplectic integration of a Poisson manifold $(M,\Lambda)$ is a symplectic groupoid $(\Gamma,\eta)$ which realizes the given Poisson manifold, i.e. such that the space of units $\Gamma_0$ with the induced Poisson structure $\Lambda_0$ is isomorphic to $(M,\Lambda)$. This notion was introduced by A. Weinstein in order to quantize Poisson manifolds by quantizing their symplectic integration. Any Poisson manifold can be integrated by a local symplectic groupoid but already for regular Poisson manifolds there are obstructions to global integrability. The aim of this paper is to summarize all the known obstructions and present a sufficient topological condition for integrability of regular Poisson manifolds; we will indeed describe a concrete procedure for this integration. Further our criterion will provide necessary and sufficient if we require $\Gamma$ to be Hausdorff, which is a suitable condition to proceed to Weinstein's program of quantization. These integrability results may be interpreted as an generalization of the Cartan-Smith proof of Lie's third theorem in the infinite dimensional case.