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The Poincaré lemma and local embeddability

Judith BrinkschulteC. Denson HillMauro Nacinovich — 2003

Bollettino dell'Unione Matematica Italiana

For pseudocomplex abstract C R manifolds, the validity of the Poincaré Lemma for 0 , 1 forms implies local embeddability in C N . The two properties are equivalent for hypersurfaces of real dimension 5 . As a corollary we obtain a criterion for the non validity of the Poicaré Lemma for 0 , 1 forms for a large class of abstract C R manifolds of C R codimension larger than one.

Obstructions to generic embeddings

Judith BrinkschulteC. Denson HillMauro Nacinovich — 2002

Annales de l’institut Fourier

Let F be a relatively closed subset of a Stein manifold. We prove that the ¯ -cohomology groups of Whitney forms on F and of currents supported on F are either zero or infinite dimensional. This yields obstructions of the existence of a generic C R embedding of a CR manifold M into any open subset of any Stein manifold, namely by the nonvanishing but finite dimensionality of some intermediate ¯ M -cohomology groups.

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