The Poincare lemma and local embeddability

TL;DR

This paper explores the relationship between the Poincare Lemma and local embeddability in CR manifolds, establishing equivalences and criteria for non-validity in higher codimension cases.

Contribution

It proves the equivalence of the Poincare Lemma and local embeddability for certain CR manifolds and provides a criterion for when the Poincare Lemma fails in higher codimension.

Findings

Poincare Lemma validity implies local embeddability for pseudoconvex CR manifolds.

Equivalence of the Poincare Lemma and local embeddability for hypersurfaces of dimension ≥ 5.

A criterion for non-validity of the Poincare Lemma in higher codimension CR manifolds.

Abstract

For pseudoconvex abstract CR manifolds, the validity of the Poincare Lemma for (0,1) forms implies local embeddability in C^N. The two properties are equivalent for hypersurfaces of real dimension > or = 5. As a corollary we obtain a criterion for the non validity of the Poicare Lemma for (0,1) forms for a large class of abstract CR manifolds of CR codimension larger than one.