Problème du bord dans l'espace projectif complexe
Annales de l'Institut Fourier, Volume 48 (1998) no. 5, p. 1483-1512

We prove that a maximally complex real submanifold Γ of dimension 2p-1 in an (n-p+1)-linearly concave open set X of n is the boundary of the analytic subset of dimension p of XΓ if and only if there exists a (p-2)-generic subset V of X * such that, for every νV the intersection Γ ν n-p+1 is the boundary of a Riemann surface (for p=2, V is O-generic if and only if it is not included in an countable union of hyperplanes of n* ). This theorem generalizes Wermer-Harvey-Lawson theorem and Dolbeault-Henkin theorem. We deduce a generalized Hartogs-Levi theorem, the theorem of extension of CR-meromorphic functions and a necessary and sufficient condition for an analytic subset of pure dimension p2 of n , to be algebraic.

Nous démontrons qu’une sous-variété réelle Γ de dimension 2p-1 et maximalement complexe d’un ouvert (n-p+1)-linéairement concave X de n est le bord d’un sous-ensemble analytique de dimension p de XΓ si et seulement s’il existe un sous-ensemble (p-2)-générique V de X * tel que pour tout νV l’intersection Γ ν n-p+1 soit le bord d’une surface de Riemann (pour p=2, V est 0-générique si et seulement s’il n’est pas inclus dans une réunion dénombrable d’hyperplans de n* ). Ce théorème généralise le théorème de Wermer-Harvey-Lawson et le théorème de Dolbeault-Henkin. Nous en déduisons le théorème de Hartogs-Levi généralisé, le théorème d’extension des fonctions CR-méromorphes et une condition nécessaire et suffisante pour qu’un sous-ensemble analytique de dimension pure p2 de n soit algébrique.

@article{AIF_1998__48_5_1483_0,
     author = {Dinh, Tien-Cuong},
     title = {Probl\`eme du bord dans l'espace projectif complexe},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {48},
     number = {5},
     year = {1998},
     pages = {1483-1512},
     doi = {10.5802/aif.1663},
     zbl = {0916.32011},
     mrnumber = {99m:32010},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1998__48_5_1483_0}
}
Dinh, Tien-Cuong. Problème du bord dans l'espace projectif complexe. Annales de l'Institut Fourier, Volume 48 (1998) no. 5, pp. 1483-1512. doi : 10.5802/aif.1663. http://www.numdam.org/item/AIF_1998__48_5_1483_0/

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