Formal equivalence of Poisson structures around Poisson submanifolds

TL;DR

This paper establishes conditions under which Poisson structures are formally equivalent around Poisson submanifolds, extending normal form results for symplectic leaves through cohomological vanishing criteria.

Contribution

It proves formal rigidity of Poisson structures around submanifolds under certain cohomological conditions and provides a formal normal form theorem around symplectic leaves.

Findings

Vanishing of specific cohomology groups implies formal rigidity.

Criteria identified for cohomology vanishing around symplectic leaves.

Established a formal version of the normal form theorem for Poisson manifolds.

Abstract

Let (M, {\pi} ) be a Poisson manifold. A Poisson submanifold $P \in M$ gives rise to an algebroid $AP \rightarrow P$, to which we associate certain chomology groups which control formal deformations of {\pi} around P . Assuming that these groups vanish, we prove that {\pi} is formally rigid around P , i.e. any other Poisson structure on M , with the same first order jet along P as {\pi} is formally Poisson diffeomorphic to {\pi} . When P is a symplectic leaf, we find a list of criteria which imply that these cohomological obstructions vanish. In particular we obtain a formal version of the normal form theorem for Poisson manifolds around symplectic leaves.