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The Poincaré lemma and local embeddability
For pseudocomplex abstract manifolds, the validity of the Poincaré Lemma for forms implies local embeddability in . The two properties are equivalent for hypersurfaces of real dimension . As a corollary we obtain a criterion for the non validity of the Poicaré Lemma for forms for a large class of abstract manifolds of codimension larger than one.
Obstructions to generic embeddings
Let be a relatively closed subset of a Stein manifold. We prove that the -cohomology groups of Whitney forms on and of currents supported on are either zero or infinite dimensional. This yields obstructions of the existence of a generic embedding of a CR manifold into any open subset of any Stein manifold, namely by the nonvanishing but finite dimensionality of some intermediate -cohomology groups.
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