Rigidity of actions on presymplectic manifolds

TL;DR

This paper establishes the rigidity of presymplectic Lie algebra actions on manifolds of constant rank, employing a novel smoothing operator and an abstract normal form theorem with Nash-Moser iteration.

Contribution

It introduces a new smoothing operator for differential forms and multivector fields, enabling the proof of rigidity for presymplectic actions using an abstract normal form theorem.

Findings

Rigidity of presymplectic actions proven

Development of a new smoothing operator for presymplectic forms

Application of Nash-Moser iteration in this context

Abstract

We prove the rigidity of presymplectic actions of a compact semisimple Lie algebra on a presymplectic manifold of constant rank in the local and global case. The proof uses an abstract normal form theorem we had stated in a previous work, based on an iterative process of Nash-Moser type. In order to use correctly this abstract theorem, we need to construct a new smoothing operator for differential forms and multivector fields which preserves the presymplectic feature.