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SEMIGLOBAL EXTENSION OF MAXIMALLY COMPLEX SUBMANIFOLDS

Part of: CR manifolds

Published online by Cambridge University Press:  13 July 2011

GIUSEPPE DELLA SALA  and
ALBERTO SARACCO
GIUSEPPE DELLA SALA
Affiliation:
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Wien, Austria (email: [email protected])
ALBERTO SARACCO*
Affiliation:
Dipartimento di Matematica, Università di Parma, viale G. Usberti 53/A, I-43124 Parma, Italy (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let A be a domain of the boundary of a (weakly) pseudoconvex domain Ω of ℂn and M a smooth, closed, maximally complex submanifold of A. We find a subdomain E of ℂn, depending only on Ω and A, and a complex variety W⊂E such that bW=M in E. Moreover, a generalization to analytic sets of depth at least 4 is given.

Keywords

boundary problempseudoconvexitymaximally complex submanifoldCR geometry

MSC classification

Secondary: 32V25: Extension of functions and other analytic objects from CR manifolds
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 3 , December 2011 , pp. 458 - 474
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

Giuseppe Della Sala has been partially supported by the BMWF grant Y377, Biholomorphic Equivalence: Analysis, Algebra and Geometry and Alberto Saracco has been supported by the MIUR Project Geometric Properties of Real and Complex Manifolds and by GNSAGA of INdAM.

References

[1]Alexander, H. and Stout, E. L., ‘A note on hulls’, Bull. Lond. Math. Soc. 22(3) (1990), 258–260.CrossRefGoogle Scholar
[2]Andreotti, A. and Siu, Y.-T., ‘Projective embedding of pseudoconcave spaces’, Ann. Sc. Norm. Super. Pisa (3) 24 (1970), 231–278.Google Scholar
[3]Chirka, E. M., Complex Analytic Sets, Mathematics and its Applications (Soviet Series), 46 (Kluwer, Dordrecht, 1989), Translated from the Russian by R. A. M. Hoksbergen.CrossRefGoogle Scholar
[4]Della Sala, G. and Saracco, A., ‘Non compact boundaries of complex analytic varieties’, Internat. J. Math. 18(2) (2007), 203–218.CrossRefGoogle Scholar
[5]Dinh, T.-C., ‘Conjecture de Globevnik–Stout et théorème de Morera pur une chaǐne holomorphe’, Ann. Fac. Sci. Toulouse Math. (6) 8 (1999), 235–257.CrossRefGoogle Scholar
[6]Dolbeault, P. and Henkin, G., ‘Surfaces de Riemann de bord donné dans CPn’, in: Contributions to Complex Analysis and Analytic Geometry, Aspects of Mathematics, E26 (Vieweg, Braunschweig, 1994), pp. 163–187.Google Scholar
[7]Dolbeault, P. and Henkin, G., ‘Chaǐnes holomorphes de bord donné dans CPn’, Bull. Soc. Math. France 125 (1997), 383–445.CrossRefGoogle Scholar
[8]Federer, H., Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, 153 (Springer, New York, 1969).Google Scholar
[9]Harvey, F. R. and Lawson, H. B. Jr, ‘On boundaries of complex analytic varieties. I’, Ann. of Math. (2) 102 (1975), 223–290.CrossRefGoogle Scholar
[10]Harvey, F. R. and Lawson, H. B. Jr, ‘Addendum to Theorem 10.4 in “Boundaries of analytic varieties”’ (2000), arXiv:math.CV/0002195.Google Scholar
[11]Harvey, F. R. and Lawson, H. B. Jr, ‘Boundaries of varieties in projective manifolds’, J. Geom. Anal. 14 (2004), 673–695.CrossRefGoogle Scholar
[12]Harvey, F. R. and Lawson, H. B. Jr, ‘Projective linking and boundaries of positive holomorphic chains in projective manifolds, part I’, in: The Many Facets of Geometry (Oxford University Press, Oxford, 2010), pp. 261–280.CrossRefGoogle Scholar
[13]Harvey, F. R. and Lawson, H. B. Jr, ‘Boundaries of positive holomorphic chains and the relative Hodge question’, Astérisque 328 (2010), 207–221.Google Scholar
[14]Harvey, F. R. and Lawson, H. B. Jr, Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds, Part II, Nankai Tracts in Mathematics, 11 (World Scientific, Hackensack, NJ, 2006), pp. 365–380.Google Scholar
[15]Lewy, H., ‘On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables’, Ann. of Math. (2) 64 (1956), 514–522.CrossRefGoogle Scholar
[16]Lupacciolu, G., ‘A theorem on holomorphic extension of CR-functions’, Pacific J. Math. 124 (1986), 177–191.CrossRefGoogle Scholar
[17]Saracco, A. and Tomassini, G., ‘Cohomology and extension problems for semi q-coronae’, Math. Z. 256(4) (2007), 737–748.CrossRefGoogle Scholar
[18]Saracco, A. and Tomassini, G., ‘Cohomology of semi 1-coronae and extension of analytic subsets’, Bull. Sci. Math. 132(3) (2008), 232–245.CrossRefGoogle Scholar
[19]Tomassini, G., ‘Sur les algèbres et d’un domaine pseudoconvexe non borné’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 10 (1983), 243–256.Google Scholar
[20]Wermer, J., ‘The hull of a curve in C n’, Ann. of Math. (2) 68 (1958), 550–561.CrossRefGoogle Scholar