Problème du bord dans l'espace projectif complexe

Dinh, Tien-Cuong

Annales de l'Institut Fourier, Tome 48 (1998), p. 1483-1512 / Harvested from Numdam

Résumé

Nous démontrons qu’une sous-variété réelle $\Gamma $ de dimension $2p-1$ et maximalement complexe d’un ouvert $(n-p+1)$-linéairement concave $X$ de ${\Bbb C}{\Bbb P}^n$ est le bord d’un sous-ensemble analytique de dimension $p$ de $X\setminus \Gamma $ si et seulement s’il existe un sous-ensemble $(p-2)$-générique $V$ de $X^*$ tel que pour tout $\nu \in V$ l’intersection $\Gamma \cap {\Bbb P}^{n-p+1}_\nu $ 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 ${\Bbb C}{\Bbb P}^{n*}$). 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 $p\ge 2$ de ${\Bbb C}^n$ soit algébrique.

Abstract

We prove that a maximally complex real submanifold $\Gamma $ of dimension $2p-1$ in an $(n-p+1)$-linearly concave open set $X$ of ${\Bbb C}{\Bbb P}^n$ is the boundary of the analytic subset of dimension $p$ of $X\setminus \Gamma $ if and only if there exists a $(p-2)$-generic subset $V$ of $X^*$ such that, for every $\nu \in V$ the intersection $\Gamma \cap {\Bbb P}^{n-p+1}_\nu $ 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 ${\Bbb C}{\Bbb P}^{n*}$). 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 $p\ge 2$ of ${\Bbb C}^n$, to be algebraic.

Détail

Publié le : 1998-01-01

MR 99m:32010 | Zbl 0916.32011 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1663

Bibliographie

[1] H. Alexander, Polynomial approximation and hulls in sets of finite linear measure in ℂn, Amer. J. Math., 93 (1971), 65-74. | MR 44 #1841 | Zbl 0221.32011

[2] E. Bishop, Subalgebras of functions on a Riemann surface, Pacific J. Math., 8 (1958), 29-50. | MR 20 #3300 | Zbl 0083.06201

[3] E. M. Chirka, Complex Analytic Sets, Kluwer Academic Publishers, 1986.

[4] T. C. Dinh, Chaînes holomorphes à bord rectifiable, C. R. Acad. Sci. Paris, 322, Série I (1996), 1135-1140. | MR 97e:32007 | Zbl 0865.32007

[5] T. C. Dinh, Enveloppe polynomiale d'un compact de longueur finie et chaînes holomorphes à bord rectifiable, Acta Mathematica, 180-1 (1998), 31-67. | MR 99f:32016 | Zbl 0942.32008

[6] T. C. Dinh, Orthogonal measures on the boundary of a Riemann surface and polynomial hull of compacts of finite length, J. Funct. Analysis, 157 (1998), 624-649. | MR 99h:32013 | Zbl 0951.32005

[7] P. Dolbeault et G. Henkin, Surfaces de Riemann de bord donné dans ℂℙn, Aspects of Mathematics, 26, Vieweg (1994), 163-187. | MR 96a:32020 | Zbl 0821.32008

[8] P. Dolbeault et G. Henkin, Chaînes holomorphes de bord donné dans ℂℙn, Bull. Soc. Math. de France, 125 (1997), 383-445. | Numdam | MR 98m:32014 | Zbl 0942.32007

[9] P. Dolbeault et J. B. Poly, Variations sur le problème des bords dans ℂℙn, Prépublication, (1995).

[10] F. Federer, Geometric Measure Theory, Grundlenhren der Math. Wiss, 285, Springer, Berlin-Heidelberg-New York, 1988.

[11] L. Gruman, The area of analytic varieties in ℂn, Math. Scand., 41 (1977), 365-379. | MR 57 #16670 | Zbl 0376.32008

[12] L. Gruman, La géométrie globale des ensembles analytiques dans ℂn, Séminaire Lelong-Skoda 1978/1979. | Zbl 0446.32007

[13] R. Harvey, Holomorphic chains and their boundaries, Proc. Symp. Pure Math., 30, vol. 1 (1977), 309-382. | MR 56 #5929 | Zbl 0374.32002

[14] R. Harvey and B. Lawson, On boundaries of complex analytic varieties I, Ann. of Math., 102 (1975), 233-290. | MR 54 #13130 | Zbl 0317.32017

[15] R. Harvey and B. Lawson, On boundaries of complex analytic varieties II, Ann. of Math., 106 (1977), 213-238. | MR 58 #17186 | Zbl 0361.32010

[16] R. Harvey and B. ShiffmanA characterization of holomorphic chains, Ann. of Math. (2), 99 (1974), 553-587. | MR 50 #7572 | Zbl 0287.32008

[17] G. Henkin, The Abel-Radon transform and several complex variables, Ann. of Math. Stud., 7 (1995), 223-275. | MR 96k:32014 | Zbl 0848.32012

[18] J. King, The currents defined by analytic varieties, Acta Math., 127 (1971), 185-220. | MR 52 #14359 | Zbl 0224.32008

[19] M. G. Lawrence, Polynomial hulls of rectifiable curves, Amer. J. Math., 117 (1995), 405-417. | MR 96d:32012 | Zbl 0827.32012

[20] M. G. Lawrence, Polynomial hulls of sets of finite length in strictly convex boundaries, Manuscript (1997).

[21] R. E. Molzon, B. Shiffman and N. Sibony, Average growth estimates for hyperplane sections of entire analytic sets, Math. Ann., 257 (1981), 43-59. | MR 83g:32008 | Zbl 0537.32009

[22] F. Sarkis, CR-meromorphic extension and the non-embedding of the Andreotti-Rossi CR structure in the projective space, Prépublication de Paris VI, 116 (1997). | Zbl 1110.32308

[23] G. Stolzenberg, Uniform approximation on smooth curves, Acta Math., 115 (1966), 185-198. | MR 33 #307 | Zbl 0143.30005

[24] J. Wermer, The hull of a curve in ℂn, Ann. of Math., 68 (1958), 550-561. | MR 20 #6536 | Zbl 0084.33402

[25] J. Wermer, Function rings and Riemann surfaces, Ann. of Math., 67 (1958), 45-71. | MR 20 #109 | Zbl 0081.32902