Arkiv för Matematik

  • Ark. Mat.
  • Volume 41, Number 1 (2003), 151-163.

Hartogs-Bochner type theorem in projective space

Frédéric Sarkis

Full-text: Open access

Abstract

We prove the following Hartogs-Bochner type theorem: Let M be a connected C2 hypersurface of Pn(C) (n≥2) which divides Pn(C) in two connected open sets Ω1 and Ω2. Suppose that M has at most one open CR orbit. Then there exists i∈{1,2} such that C1 CR functions defined on M extends holomorphically to Ωi.

Note

Supported by the TMR network.

Article information

Source
Ark. Mat., Volume 41, Number 1 (2003), 151-163.

Dates
Received: 3 December 2000
Revised: 16 April 2002
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898797

Digital Object Identifier
doi:10.1007/BF02384573

Mathematical Reviews number (MathSciNet)
MR1971946

Zentralblatt MATH identifier
1039.32044

Rights
2003 © Institut Mittag-Leffler

Citation

Sarkis, Frédéric. Hartogs-Bochner type theorem in projective space. Ark. Mat. 41 (2003), no. 1, 151--163. doi:10.1007/BF02384573. https://projecteuclid.org/euclid.afm/1485898797


Export citation

References

  • Chirka, E. M., Complex Analytic Sets, Nauka, Moscow, 1985 (Russian). English transl.: Kluwer, Dordrecht, 1989.
  • Dinh, T. C., Enveloppe polynomiale d’un compact de longueur finie et chaînes holomorphes à bord rectifiable, Acta Math. 180 (1998), 31–67.
  • Dolbeault, P. and Henkin, G., Chaînes holomorphes de bord donné dans un ouvert q-concave de C Pn. Bull. Soc. Math. France 125 (1997), 383–445.
  • Dwilewicz, R. and Merker, J., On the Hartogs-Bochner phenomenon for CR functions in P2(C), Proc. Amer. Math. Soc. 130 (2002), 1975–1980.
  • Fujita, R., Domaines sans point critiques intérieur sur l’espace projectif complexe, J. Math. Soc. Japan 15 (1963), 443–473.
  • Fujita, R., Domaines sans point critiques intérieur sur l’espace produit. J. Math. Kyoto Univ. 4 (1965), 493–514.
  • Ghys, É., Laminations par surfaces de Riemann, in Dynamique et Géométrie Complexes (Lyon, 1997) (Cerveau, D., Ghys, É., Sibony, N. and Yoccoz, J.-C., authors), Panoramas et Synthèses 8, pp. ix, xi, 49–95, Soc. Math. France, Paris, 1999.
  • Hartogs, F., Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen, Münch. Ber. 36 (1906), 223–242.
  • Harvey, R. and Lawson, B., On boundaries of complex analytic varieties, I, Ann. of Math. 102 (1975), 233–290.
  • Henkin, G. and Iordan, A., Regularity of $\bar \partial $ on pseudoconcave compacts and applications, Asian J. Math. 4 (2000), 855–883.
  • Jöricke, B., Some remarks concerning holomorphically convex hulls and envelopes of holomorphy, Math. Z. 218 (1995), 143–157.
  • Jöricke, B., Boundaries of singularity sets, removable singularities, and CR invariant subsets of CR manifolds, J. Geom. Anal. 9 (1999), 257–300.
  • Kiselman, C. O., On entire functions of exponential type and indicators of analytic functions, Acta Math. 117 (1967), 1–35.
  • Koziarz, V. and Sarkis, F., Problème du bord dans les variétés q-convexes et phénomène de Hartogs-Bochner, Math. Ann. 321 (2000), 569–585.
  • Merker, J., Global minimality of generic manifolds and holomorphic extendibility of CR functions, Internat. Math. Res. Notices (1994), 329–343.
  • Porten, E., A Hartogs-Bochner type theorem for continuous CR mappings, Manuscript, 1996.
  • Sarkis, F., CR meromorphic extension and the non-embedding of the Andreotti-Rossi CR structure in the projective space. Internat. J. Math. 10 (1999), 897–915.
  • Sarkis, F., Problème Plateau complexe dans les variétés Kählériennes, Bull. Soc. Math. France 130 (2002), 169–209.
  • Sussmann, H. J., Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–187.
  • Takeuchi, A., Domaines pseudoconvexes infinis et la riemannienne dans un espace projectif, J. Math. Soc. Japan 16 (1964), 159–181.
  • Trépreau, J. M., Sur la propagation des singularités dans les variétés CR, Bull. Soc. Math. France 118 (1990), 403–450.