Obstructions to generic embeddings
Annales de l'Institut Fourier, Volume 52 (2002) no. 6, pp. 1785-1792.

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 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 ¯ M -cohomology groups.

Soit F un ensemble relativement fermé d’une variété de Stein. On prouve que les groupes de cohomologie associés à l’opérateur ¯ 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 ¯ M soient ou bien zéro ou bien de dimension infinie.

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 $CR$
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)
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Brinkschulte, Judith; Denson Hill, C.; Nacinovich, Mauro. Obstructions to generic embeddings. Annales de l'Institut Fourier, Volume 52 (2002) no. 6, pp. 1785-1792. doi : 10.5802/aif.1934. https://aif.centre-mersenne.org/articles/10.5802/aif.1934/

[AFN] A. Andreotti; G. Fredricks; M. Nacinovich On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. Sc. Norm. Sup. Pisa, Volume 8 (1981), pp. 365-404 | Numdam | MR | Zbl

[AH] A. Andreotti; C.D. Hill Complex characteristic coordinates and tangential Cauchy-Riemann equations, Ann. Sc. Norm. Sup. Pisa, Volume 26 (1972), pp. 299-324 | Numdam | MR | Zbl

[AHLM] A. Andreotti; C.D. Hill; S. Lojasiewicz; B. MacKichan Complexes of differential operators. The Mayer-Vietoris sequence, Invent. Math, Volume 35 (1976), pp. 43-86 | MR | Zbl

[B] G.E. Bredon Sheaf theory, GTM, 170, Springer-Verlag, 1997 | MR | Zbl

[Br] J. Brinkschulte Laufer's vanishing theorem for embedded CR manifolds, Math. Z, Volume 239 (2002), pp. 863-866 | DOI | MR | Zbl

[G] H. Grauert On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math, Volume 68 (1958), pp. 460-472 | DOI | MR | Zbl

[HL] R. Harvey; L.B. Lawson On the boundaries of complex analytic varieties I, Ann. of Math, Volume 102 (1975), pp. 223-290 | DOI | MR | Zbl

[HN1] C.D. Hill; M. Nacinovich A necessary condition for global Stein immersion of compact CR manifolds, Riv. Mat. Univ. Parma, Volume 5 (1992), pp. 175-182 | MR | Zbl

[HN2] C.D. Hill; M. Nacinovich Duality and distribution cohomology of CR manifolds, Ann. Sc. Norm. Sup. Pisa, Volume 22 (1995), pp. 315-339 | Numdam | MR | Zbl

[L] H.B. Laufer On the infinite dimensionality of the Dolbeault cohomology groups, Proc. Amer. Math. Soc, Volume 52 (1975), pp. 293-296 | DOI | MR | Zbl

[N1] M. Nacinovich On boundary Hilbert differential complexes, Ann. Polon. Math, Volume 46 (1985), pp. 213-235 | MR | Zbl

[N2] M. Nacinovich Poincaré lemma for tangential Cauchy-Riemann complexes, Math. Ann, Volume 268 (1984), pp. 449-471 | DOI | MR | Zbl

[NV] M. Nacinovich; G. Valli Tangential Cauchy-Riemann complexes on distributions, Ann. Mat. Pura Appl, Volume 146 (1987), pp. 123-160 | DOI | MR | Zbl

[Y] S.-T. Yau Kohn-Rossi cohomology and its application to the complex Plateau problem I, Ann. of Math, Volume 113 (1981), pp. 67-110 | DOI | MR | Zbl

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