Obstructions to generic embeddings

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

doi:10.5802/aif.1934

Obstructions to generic embeddings
[Obstructions aux plongements génériques]

Brinkschulte, Judith ¹ ; Denson Hill, C.² ; Nacinovich, Mauro ³

¹ Chalmers University of Technology \& Göteborg University, Department of Mathematics, Göteborg (Suède)
² SUNY at Stony Brook, Department of Mathematics, Stony Brook NY 11794 (USA)
³ Università di Roma "Tor Vergaga", Dipartimento di Matematica, Via della Ricerca Scientifica, 00133 Roma (Italie)

Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1785-1792.

Résumé
Abstract

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 $C R$ 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.

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 $C R$ embedding of a CR manifold $M$ into any open subset of any Stein manifold, namely by the nonvanishing but finite dimensionality of some intermediate ${\bar{\partial}}_{M}$ -cohomology groups.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1934

Classification : 32V05, 32V30
Keywords: $\bar{\partial }$-operator, tangential $CR$ operator, embedding of $CR$ manifolds
Mot clés : $\bar{\partial }$-opérateur, opérateur $CR$ tangentiel, plongement de variétés $CR$

Affiliations des auteurs :

Brinkschulte, Judith ¹ ; Denson Hill, C. ² ; Nacinovich, Mauro ³

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     title = {Obstructions to generic embeddings},
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Brinkschulte, Judith; Denson Hill, C.; Nacinovich, Mauro. Obstructions to generic embeddings. Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1785-1792. doi : 10.5802/aif.1934. http://www.numdam.org/articles/10.5802/aif.1934/

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