On the Hartogs–Bochner phenomenon for CR functions in $P_2(\mathbb {C})$
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- by Roman Dwilewicz and Joël Merker PDF
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Abstract:
Let $M$ be a compact, connected, $\mathcal {C}^2$-smooth and globally minimal hypersurface $M$ in $P_2(\mathbb {C})$ which divides the projective space into two connected parts $U^{+}$ and $U^{-}$. We prove that there exists a side, $U^-$ or $U^+$, such that every continuous CR function on $M$ extends holomorphically to this side. Our proof of this theorem is a simplification of a result originally due to F. Sarkis.References
- Dominique Cerveau, Minimaux des feuilletages algébriques de $\textbf {C}\textrm {P}(n)$, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1535–1543 (French, with English and French summaries). MR 1275208
- Pierre Dolbeault and Gennadi Henkin, Chaînes holomorphes de bord donné dans $\textbf {C}\textrm {P}^n$, Bull. Soc. Math. France 125 (1997), no. 3, 383–445 (French, with English and French summaries). MR 1605457
- Leon Ehrenpreis, A new proof and an extension of Hartogs’ theorem, Bull. Amer. Math. Soc. 67 (1961), 507–509. MR 131663, DOI 10.1090/S0002-9904-1961-10661-7
- Bruno Fabre, Sur l’intersection d’une surface de Riemann avec des hypersurfaces algébriques, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 4, 371–376 (French, with English and French summaries). MR 1378515
- Étienne Ghys, Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997) Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. ix, xi, 49–95 (French, with English and French summaries). MR 1760843
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523, DOI 10.1002/9781118032527
- Reese Harvey, Holomorphic chains and their boundaries, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975) Amer. Math. Soc., Providence, R.I., 1977, pp. 309–382. MR 0447619
- Howard Levi, On the values assumed by polynomials, Bull. Amer. Math. Soc. 45 (1939), 570–575. MR 54, DOI 10.1090/S0002-9904-1939-07038-9
- G. M. Henkin and J. Leiterer, Theory of functions on complex manifolds, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], vol. 60, Akademie-Verlag, Berlin, 1984. MR 795028
- S. M. Ivashkovich, The Hartogs-type extension theorem for meromorphic maps into compact Kähler manifolds, Invent. Math. 109 (1992), no. 1, 47–54. MR 1168365, DOI 10.1007/BF01232018
- Burglind Jöricke, Some remarks concerning holomorphically convex hulls and envelopes of holomorphy, Math. Z. 218 (1995), no. 1, 143–157. MR 1312583, DOI 10.1007/BF02571894
- J. J. Kohn and Hugo Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. (2) 81 (1965), 451–472. MR 177135, DOI 10.2307/1970624
- Christine Laurent-Thiébaut, Phénomène de Hartogs-Bochner dans les variétés CR, Topics in complex analysis (Warsaw, 1992) Banach Center Publ., vol. 31, Polish Acad. Sci. Inst. Math., Warsaw, 1995, pp. 233–247 (French). MR 1341392
- Alcides Lins Neto, A note on projective Levi flats and minimal sets of algebraic foliations, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 4, 1369–1385 (English, with English and French summaries). MR 1703092
- J. Merker, Global minimality of generic manifolds and holomorphic extendibility of CR functions, Internat. Math. Res. Notices 8 (1994), 329 ff., approx. 14 pp.}, issn=1073-7928, review= MR 1289578, doi=10.1155/S1073792894000383,
- Frédéric Sarkis, CR-meromorphic extension and the nonembeddability of the Andreotti-Rossi CR structure in the projective space, Internat. J. Math. 10 (1999), no. 7, 897–915. MR 1728127, DOI 10.1142/S0129167X99000380
- F. Sarkis, Problème de Plateau complexe dans les variétés kahlériennes. Preprint, 1999.
- Y.T. Siu, Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension $\geq 3$. Ann. Math. (2) 151 (2000), no. 3, 1217–1243.
- Akira Takeuchi, Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif, J. Math. Soc. Japan 16 (1964), 159–181 (French). MR 173789, DOI 10.2969/jmsj/01620109
- J.-M. Trépreau, Sur la propagation des singularités dans les variétés CR, Bull. Soc. Math. France 118 (1990), no. 4, 403–450 (French, with English summary). MR 1090408
Additional Information
- Roman Dwilewicz
- Affiliation: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, P.O. Box 137, 00-950 Warsaw, Poland
- Email: rd@impan.gov.pl
- Joël Merker
- Affiliation: Laboratoire d’Analyse, Topologie et Probabilités, Centre de Mathématiques et Informatique, UMR 6632, 39 rue Joliot Curie, F-13453 Marseille Cedex 13, France
- Email: merker@cmi.univ-mrs.fr
- Received by editor(s): December 13, 2000
- Published electronically: February 27, 2002
- Additional Notes: This research was partially supported by a grant of the Polish Committee for Scientific Research KBN 2 PO3A 044 15 and by a grant from the French-Polish program “Polonium 1999”
- Communicated by: Steven R. Bell
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1975-1980
- MSC (2000): Primary 32V25; Secondary 32V10, 32V15, 32D15
- DOI: https://doi.org/10.1090/S0002-9939-02-06357-8
- MathSciNet review: 1896029