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Stability of embeddings for pseudoconcave surfaces and their boundaries

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References

  1. Andreotti, A., Théorèmes de dépendence algébrique sur les espaces complexes pseudoconcave.Bull. Soc. Math. France, 91 (1963), 1–38.

    MATH  MathSciNet  Google Scholar 

  2. Andreotti, A. &Siu, Y.-T., Projective embedding of pseudoconcave spaces.Ann. Scuola Norm. Sup. Pisa (3), 24 (1970), 231–278.

    MathSciNet  MATH  Google Scholar 

  3. Andreotti, A. &Tomassini, G., Some remarks on pseudoconcave manifolds, inEssays on Topology and Related Topics (mémoires dédiés à Georges de Rham), pp. 85–104. Springer-Verlag, New York, 1970.

    Google Scholar 

  4. Bishop, E., Conditions for the analyticity of certain sets.Michigan Math. J., 11 (1964), 289–304.

    Article  MATH  MathSciNet  Google Scholar 

  5. Bland, J., Contact geometry and CR structures onS 3.Acta Math., 172 (1994), 1–49.

    MATH  MathSciNet  Google Scholar 

  6. Bland, J. &Duchamp, T., Deformation theory for the hyperplane line bundle on P1, inCR-Geometry and Overdetermined Systems (Osaka, 1994), pp. 41–59. Adv. Stud. Pure Math., 25. Math. Soc. Japan, Tokyo, 1997.

    Google Scholar 

  7. Bland, J. &Epstein, C. L., Embeddable CR-structures and deformations of pseudoconvex surfaces, I. Formal deformations.J. Algebraic Geom., 5 (1996), 277–368.

    MathSciNet  MATH  Google Scholar 

  8. Bogomolov, F., On fillability of contact structures on 3-dimensional manifolds. Preprint, Göttingen, 1993.

  9. Bott, R., Homogeneous vector bundles.Ann. of Math., 66 (1957), 203–248.

    Article  MATH  MathSciNet  Google Scholar 

  10. Barth, W., Peters, C. &Van de Ven, A.,Compact Complex Surfaces. Ergeb. Math. Grenzgeb. (3), 4. Springer-Verlag, Berlin-New York, 1984.

    MATH  Google Scholar 

  11. Burns, D. M. &Epstein, C. L., Embeddability for three-dimensional CR-manifolds.J. Amer. Math. Soc., 3 (1990), 809–841.

    Article  MathSciNet  MATH  Google Scholar 

  12. Chow, W.-L. &Kodaira, K., On analytic surfaces with two independent meromorphic functions.Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 319–325.

    MathSciNet  MATH  Google Scholar 

  13. Catlin, D. &Lempert, L., A note on the instabilityof embeddings of Cauchy-Riemann manifolds.J. Geom. Anal., 2 (1992), 99–104.

    MathSciNet  MATH  Google Scholar 

  14. Docquier, F. &Grauert, H., Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten.Math. Ann., 140 (1960), 94–123.

    Article  MathSciNet  MATH  Google Scholar 

  15. Epstein, C. L. &Henkin, G. M., Extension of CR-structures for 3-dimensional pseudoconcave manifolds, inMultidimensional Complex Analysis and Partial Differential Equations (São Carlos, 1995), pp. 51–67. Contemp. Math., 205. Amer. Math. Soc., Providence, RI, 1997.

    Google Scholar 

  16. —, Two lemmas in local analytic geometry, inAnalysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998), pp. 189–195. Contemp. Math., 251. Amer. Math. Soc., Providence, RI, 2000.

    Google Scholar 

  17. —, Embeddings for 3-dimensional CR-manifolds, inComplex Analysis and Geometry (Paris, 1997), pp. 223–236. Progr. Math., 188. Birkhäuser, Basel, 2000.

    Google Scholar 

  18. Eliashberg, Y., Filling by holomorphic discs and its applications, inGeometry of Low-Dimensional Manifolds, 2 (Durham, 1989), pp. 45–67. London Math. Soc. Lecture Note Ser., 151. Cambridge Univ. Press, Cambridge, 1990.

    Google Scholar 

  19. Epstein, C. L., CR-structures on three-dimensional circle bundles.Invent. Math., 109 (1992), 351–403.

    Article  MATH  MathSciNet  Google Scholar 

  20. —, A relative index for embeddable CR-structures, I; II.Ann. of Math., 147 (1998), 1–59; 61–91.

    Article  MATH  MathSciNet  Google Scholar 

  21. Fabre, B., Sur l'intersection d'une surface de Riemann avec des hypersurfaces algébriques.C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 371–376.

    MATH  MathSciNet  Google Scholar 

  22. Griffiths, P. &Harris, J.,Principles of Algebraic Geometry. Wiley-Interscience, New York, 1978.

    MATH  Google Scholar 

  23. Griffiths, P., The extension problem in complex analysis, II. Embeddings with positive normal bundle.Amer. J. Math., 88 (1966), 366–446.

    MATH  MathSciNet  Google Scholar 

  24. Grauert, H. &Riemenschneider, O., Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen.Invent. Math., 11 (1970), 263–292.

    Article  MathSciNet  MATH  Google Scholar 

  25. Gunning, R. C.,Introduction to Holomorphic Functions of Several Variables, Vol. II. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1990.

    MATH  Google Scholar 

  26. Hartshorne, R.,Algebraic Geometry. Graduate Texts in Math., 52. Springer-Verlag, New York-Heidelberg, 1977.

    MATH  Google Scholar 

  27. Harvey, R., Holomorphic chains and their boundaries, inSeveral Complex Variables (Williamstown, MA, 1975), pp. 309–382. Proc. Sympos. Pure Math., 30∶1. Amer. Math. Soc., Providence, RI, 1977.

    Google Scholar 

  28. Harvey, F. R. &Lawson, H. B., Jr., On boundaries of complex analytic varieties, I.Ann. of Math., 102 (1975), 223–290.

    Article  MathSciNet  Google Scholar 

  29. Kato, T.,Perturbation Theory of Linear Operators. Grundlehren Math. Wiss., 132. Springer-Verlag, New York, 1966.

    Google Scholar 

  30. Kodaira, K., On compact complex analytic surfaces, I; II; III.Ann. of Math., 71; 77; 78 (1960; 1963; 1963), 111–152; 563–626; 1–40.

    Article  MATH  MathSciNet  Google Scholar 

  31. —, On stability of compact submanifolds of complex manifolds.Amer. J. Math., 85 (1963), 79–94.

    MATH  MathSciNet  Google Scholar 

  32. Kiremidjian, G., A direct extension method for CR-structures.Math. Ann., 242 (1979), 1–19.

    Article  MATH  MathSciNet  Google Scholar 

  33. Kronheimer, P. B. &Mrowka, T. S., Monopoles and contact structures.Invent. Math., 130 (1997), 209–255.

    Article  MathSciNet  MATH  Google Scholar 

  34. Kohn, J. J., The range of the tangential Cauchy-Riemann operator.Duke Math. J., 53 (1986), 525–545.

    Article  MATH  MathSciNet  Google Scholar 

  35. Kohn, J. J. &Rossi, H., On the extension of holomorphic functions from the boundary of a complex manifold.Ann. of Math., 81 (1965), 451–472.

    Article  MathSciNet  Google Scholar 

  36. Kuranishi, M.,Deformations of Compact Complex Manifolds. Séminaire de Mathématiques Supérieures, 39. Les Presses de l'Université de Montréal, Montreal, 1971.

    Google Scholar 

  37. Lempert, L., On three-dimensional Cauchy-Riemann manifolds.J. Amer. Math. Soc., 5 (1992), 923–969.

    Article  MATH  MathSciNet  Google Scholar 

  38. —, Embeddings of three-dimensional Cauchy-Riemann manifolds.Math. Ann., 300 (1994), 1–15.

    Article  MATH  MathSciNet  Google Scholar 

  39. —, Algebraic approximations in analytic geometry.Invent. Math., 121 (1995), 335–353.

    Article  MATH  MathSciNet  Google Scholar 

  40. Li, H.-L., The stability of embeddings of Cauchy-Riemann manifolds. Thesis, Purdue University, 1995.

  41. Morrow, J. &Rossi, H., Some general results on equivalence of embeddings, inRecent Developments in Several Complex Variables (Princeton, NJ, 1979), pp. 299–325. Ann. of Math. Stud., 100. Princeton Univ. Press, Princeton, NJ, 1981.

    Google Scholar 

  42. —, Some theorems of algebraicity for complex spaces.J. Math. Soc. Japan, 27 (1975), 167–183.

    Article  MathSciNet  MATH  Google Scholar 

  43. Ouyang, Y., Ph.D. Thesis, University of Pennsylvania, 1999.

  44. Pinkham, H. C., Deformations of cones with negative grading.J. Algebra, 30 (1974), 92–102.

    Article  MATH  MathSciNet  Google Scholar 

  45. Pardon, W. &Stern, M. A.,L 2-\(\bar \partial \)-cohomology of complex projective varieties.J. Amer. Math. Soc., 4 (1991), 603–621.

    Article  MathSciNet  MATH  Google Scholar 

  46. Rossi, H., Attaching analytic spaces to an analytic space along a pseudoconcave boundary, inProceedings of the Conference on Complex Analysis (Minneapolis, MN, 1964), pp. 242–256. Springer-Verlag, Berlin, 1965.

    Google Scholar 

  47. Siu, Y.-T., Every Stein subvariety admits a Stein neighborhood.Invent. Math., 38 (1976), 89–100.

    Article  MATH  MathSciNet  Google Scholar 

  48. Shiffman, B. &Sommese, A. J.,Vanishing Theorems on Complex Manifolds. Progr. Math., 56. Birkhäuser, Boston, Boston, MA, 1985.

    MATH  Google Scholar 

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Research supported in part by the University of Paris VI and NSF Grant DMS96-23040.

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Epstein, C.L., Henkin, G.M. Stability of embeddings for pseudoconcave surfaces and their boundaries. Acta Math. 185, 161–237 (2000). https://doi.org/10.1007/BF02392810

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