On the failure of the Poincar\'e Lemma for de-bar-sub-M II

TL;DR

This paper investigates the limitations of the Poincaré lemma for tangential Cauchy-Riemann equations on CR manifolds, extending classical nonsolvability examples and exploring geometric degeneracies.

Contribution

It provides sharp results on the failure of the Poincaré lemma for these equations, generalizing Lewy's classical example and analyzing CR structures with degeneracies.

Findings

Demonstrates sharp non-solvability results for tangential Cauchy-Riemann equations.

Generalizes Lewy's classical nonsolvability example to broader CR manifolds.

Analyzes the CR structure on the characteristic bundle with Levi form degeneracies.

Abstract

We obtain very sharp results about the lack of validity of the Poincare lemma for the tangential Cauchy Riemann equations, acting on tangential forms, tangential to a CR manifold M of general CR dimension n, and general CR codimension k. This generalizes the classical nonsolvability example of H. Lewy. We also discuss the CR structure on the characteristic bundle to M, due to certain degeneracies in the Levi form. A number of naturally geometrically occuring examples are given.