Formal equivalence of Poisson structures around Poisson submanifolds

Abstract

Let (M,π) be a Poisson manifold. A Poisson submanifold P ⊂ M gives rise to a Lie algebroid AP → P. Formal deformations of π around P are controlled by certain cohomology groups associated to AP. Assuming that these groups vanish, we prove that π is formally rigid around P; that is, any other Poisson structure on M, with the same first-order jet along P, is formally Poisson diffeomorphic to π. When P is a symplectic leaf, we find a list of criteria that are sufficient for these cohomological obstructions to vanish. In particular, we obtain a formal version of the normal form theorem for Poisson manifolds around symplectic leaves.