On a Poincare lemma for foliations

TL;DR

This paper revisits a Poincare lemma for foliated forms, computes foliated cohomology for local models of integrable systems with nondegenerate singularities, and introduces analytical tools for such systems.

Contribution

It provides a new Poincare lemma for the deformation complex of singular foliations and computes their cohomology in local models of integrable systems.

Findings

Computed foliated cohomology for local models of integrable systems with singularities

Developed analytical tools for integrable systems with nondegenerate singularities

Established a Poincare lemma for the deformation complex of singular foliations

Abstract

In this paper we revisit a Poincare lemma for foliated forms, with respect to a regular foliation, and compute the foliated cohomology for local models of integrable systems with singularities of nondegenerate type. A key point in this computation is the use of some analytical tools for integrable systems with nondegenerate singularities, including a Poincare lemma for the deformation complex associated to this singular foliation.