Poisson geometry around Poisson submanifolds

TL;DR

This paper develops a local linearization model for Poisson manifolds around submanifolds, generalizing known theorems and providing solutions to the groupoid coisotropic embedding problem.

Contribution

It introduces a first order local model for Poisson manifolds near submanifolds and establishes conditions for this model to serve as a local normal form, extending existing linearization results.

Findings

Provides a unified linearization theorem encompassing fixed points and symplectic leaves.

Offers a solution to the groupoid coisotropic embedding problem.

Generalizes known linearization theorems in Poisson geometry.

Abstract

We construct a first order local model for Poisson manifolds around a large class of Poisson submanifolds and we give conditions under which this model is a local normal form. The resulting linearization theorem includes as special cases all the known linearization theorems for fixed points and symplectic leaves. The symplectic groupoid version of these results gives a solution to the groupoid coisotropic embedding problem.