Obstructions to generic embeddings

[Obstructions aux plongements génériques]

Brinkschulte, Judith ; Denson Hill, C. ; Nacinovich, Mauro

Annales de l'Institut Fourier, Tome 52 (2002), p. 1785-1792 / Harvested from Numdam

Résumé

Soit $F$ un ensemble relativement fermé d’une variété de Stein. On prouve que les groupes de cohomologie associés à l’opérateur $\bar{\partial }$ des formes de Whitney sur $F$ et des courants à support dans $F$ sont soit zéro, soit de dimension infinie. Cela nous permet d’obtenir une condition nécessaire pour l’existence d’un plongement $CR$ générique d’une variété CR $M$ dans un ouvert d’une variété de Stein : il faut que tous les groupes de cohomologie associés à l’opérateur $\bar{\partial }_M$ soient ou bien zéro ou bien de dimension infinie.

Abstract

Let $F$ be a relatively closed subset of a Stein manifold. We prove that the $\bar{\partial }$-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 $CR$ embedding of a CR manifold $M$ into any open subset of any Stein manifold, namely by the nonvanishing but finite dimensionality of some intermediate $\bar{\partial }_M$-cohomology groups.

Détail

Publié le : 2002-01-01

MR 1952531 | Zbl 1029.32018 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1934
Classification:  32V05,  32V30
Mots clés: $\bar{\partial }$-opérateur, opérateur $CR$ tangentiel, plongement de variétés $CR$

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