Obstructions to generic embeddings  [ Obstructions aux plongements génériques ]
Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1785-1792.

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.

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.

DOI : https://doi.org/10.5802/aif.1934
Classification : 32V05,  32V30
Mots clés: ¯-opérateur, opérateur CR tangentiel, plongement de variétés CR
@article{AIF_2002__52_6_1785_0,
     author = {Brinkschulte, Judith and Denson Hill, C. and Nacinovich, Mauro},
     title = {Obstructions to generic embeddings},
     journal = {Annales de l'Institut Fourier},
     pages = {1785--1792},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {6},
     year = {2002},
     doi = {10.5802/aif.1934},
     zbl = {1029.32018},
     mrnumber = {1952531},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2002__52_6_1785_0/}
}
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. https://aif.centre-mersenne.org/item/AIF_2002__52_6_1785_0/

[AFN] A. Andreotti; G. Fredricks; M. Nacinovich On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. Sc. Norm. Sup. Pisa, Tome 8 (1981), pp. 365-404 | Numdam | MR 634855 | Zbl 0482.35061

[AH] A. Andreotti; C.D. Hill Complex characteristic coordinates and tangential Cauchy-Riemann equations, Ann. Sc. Norm. Sup. Pisa, Tome 26 (1972), pp. 299-324 | Numdam | MR 460724 | Zbl 0256.32006

[AHLM] A. Andreotti; C.D. Hill; S. Lojasiewicz; B. MacKichan Complexes of differential operators. The Mayer-Vietoris sequence, Invent. Math, Tome 35 (1976), pp. 43-86 | MR 423425 | Zbl 0332.58016

[B] G.E. Bredon Sheaf theory, GTM, Tome 170, Springer-Verlag, 1997 | MR 1481706 | Zbl 0874.55001

[Br] J. Brinkschulte Laufer's vanishing theorem for embedded $CR$ manifolds, Math. Z, Tome 239 (2002), pp. 863-866 | Article | MR 1902064 | Zbl 1008.32021

[G] H. Grauert On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math, Tome 68 (1958), pp. 460-472 | Article | MR 98847 | Zbl 0108.07804

[HL] R. Harvey; L.B. Lawson On the boundaries of complex analytic varieties I, Ann. of Math, Tome 102 (1975), pp. 223-290 | Article | MR 425173 | Zbl 0317.32017

[HN1] C.D. Hill; M. Nacinovich A necessary condition for global Stein immersion of compact $CR$ manifolds, Riv. Mat. Univ. Parma, Tome 5 (1992), pp. 175-182 | MR 1230608 | Zbl 0787.32020

[HN2] C.D. Hill; M. Nacinovich Duality and distribution cohomology of $CR$ manifolds, Ann. Sc. Norm. Sup. Pisa, Tome 22 (1995), pp. 315-339 | Numdam | MR 1354910 | Zbl 0848.32003

[L] H.B. Laufer On the infinite dimensionality of the Dolbeault cohomology groups, Proc. Amer. Math. Soc, Tome 52 (1975), pp. 293-296 | Article | MR 379887 | Zbl 0314.32008

[N2] M. Nacinovich Poincaré lemma for tangential Cauchy-Riemann complexes, Math. Ann, Tome 268 (1984), pp. 449-471 | Article | MR 841829 | Zbl 0606.58046

[N1] M. Nacinovich On boundary Hilbert differential complexes, Ann. Polon. Math, Tome 46 (1985), pp. 213-235 | Article | MR 753407 | Zbl 0574.32045

[NV] M. Nacinovich; G. Valli Tangential Cauchy-Riemann complexes on distributions, Ann. Mat. Pura Appl, Tome 146 (1987), pp. 123-160 | Article | MR 916690 | Zbl 0631.58024

[Y] S.-T. Yau Kohn-Rossi cohomology and its application to the complex Plateau problem I, Ann. of Math, Tome 113 (1981), pp. 67-110 | Article | MR 604043 | Zbl 0464.32012