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Obstructions to generic embeddings

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Obstructions to generic embeddings AN N A L E S D E L’INSTI T U T F O U R IE R ANNALES DE L’INSTITUT FOURIER Judith BRINKSCHULTE, C. DENSON HILL &Mauro NACINOVICH Obstructions to generic embeddings Tome 52, no 6 (2002), p. 1785-1792. <http://aif.cedram.org/item?id=AIF_2002__52_6_1785_0> © Association des Annales de l’institut Fourier, 2002, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ 1785 OBSTRUCTIONS TO GENERIC EMBEDDINGS by J. BRINKSCHULTE, C. DENSON HILL &#x26; M. NACINOVICH In Grauert’s paper [G] it is noted that finite dimensionality of co- homology groups sometimes implies vanishing of these cohomology groups. Later on Laufer formulated a zero or infinity law for the cohomology groups of domains in Stein manifolds. In this paper we generalize Laufer’s The- orem in [L] and its version for small domains of CR manifolds, proved in [Br], by considering Whitney cohomology on locally closed subsets and cohomology with supports for currents. With this approach we obtain a global result for CR manifolds generically embedded in a Stein manifold. Namely a necessary condition for global embedding into an open subset of a Stein manifold is that the 8M-cohomology groups must be either zero or infinite dimensional. 1. An abstract Laufer Theorem. Let X be a Stein manifold of complex dimension N. Let F be a locally closed subset of X. This means that F is a closed subset

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