Rigidity around Poisson Submanifolds

TL;DR

This paper establishes a rigidity theorem in Poisson geometry around compact submanifolds, strengthening existing linearization and normal form results, and enabling explicit computation of Poisson moduli spaces in specific cases.

Contribution

It introduces a new rigidity theorem in Poisson geometry that generalizes and strengthens classical linearization and normal form theorems using Nash-Moser techniques.

Findings

Proves a rigidity theorem around compact Poisson submanifolds.

Strengthens Conn's linearization theorem for fixed points.

Allows explicit computation of Poisson moduli spaces for spheres in duals of Lie algebras.

Abstract

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash-Moser fast convergence method. In the case of one-point submanifolds (fixed points), this immediately implies a stronger version of Conn's linearization theorem, also proving that Conn's theorem is, indeed, just a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem; another interesting case corresponds to spheres inside duals of compact semisimple Lie algebras, our result can be used to fully compute the resulting Poisson moduli space.