Acta Mathematica

, Volume 185, Issue 2, pp 161–237

Stability of embeddings for pseudoconcave surfaces and their boundaries

  • Charles L. Epstein
  • Gennadi M. Henkin
Article

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References

  1. [A]
    Andreotti, A., Théorèmes de dépendence algébrique sur les espaces complexes pseudoconcave.Bull. Soc. Math. France, 91 (1963), 1–38.MATHMathSciNetGoogle Scholar
  2. [AS]
    Andreotti, A. &Siu, Y.-T., Projective embedding of pseudoconcave spaces.Ann. Scuola Norm. Sup. Pisa (3), 24 (1970), 231–278.MathSciNetMATHGoogle Scholar
  3. [AT]
    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. [Bi]
    Bishop, E., Conditions for the analyticity of certain sets.Michigan Math. J., 11 (1964), 289–304.CrossRefMATHMathSciNetGoogle Scholar
  5. [Bl]
    Bland, J., Contact geometry and CR structures onS 3.Acta Math., 172 (1994), 1–49.MATHMathSciNetGoogle Scholar
  6. [BID]
    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. [BlE]
    Bland, J. &Epstein, C. L., Embeddable CR-structures and deformations of pseudoconvex surfaces, I. Formal deformations.J. Algebraic Geom., 5 (1996), 277–368.MathSciNetMATHGoogle Scholar
  8. [Bog]
    Bogomolov, F., On fillability of contact structures on 3-dimensional manifolds. Preprint, Göttingen, 1993.Google Scholar
  9. [Bot]
    Bott, R., Homogeneous vector bundles.Ann. of Math., 66 (1957), 203–248.CrossRefMATHMathSciNetGoogle Scholar
  10. [BPV]
    Barth, W., Peters, C. &Van de Ven, A.,Compact Complex Surfaces. Ergeb. Math. Grenzgeb. (3), 4. Springer-Verlag, Berlin-New York, 1984.MATHGoogle Scholar
  11. [BuE]
    Burns, D. M. &Epstein, C. L., Embeddability for three-dimensional CR-manifolds.J. Amer. Math. Soc., 3 (1990), 809–841.CrossRefMathSciNetMATHGoogle Scholar
  12. [CK]
    Chow, W.-L. &Kodaira, K., On analytic surfaces with two independent meromorphic functions.Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 319–325.MathSciNetMATHGoogle Scholar
  13. [CL]
    Catlin, D. &Lempert, L., A note on the instabilityof embeddings of Cauchy-Riemann manifolds.J. Geom. Anal., 2 (1992), 99–104.MathSciNetMATHGoogle Scholar
  14. [DG]
    Docquier, F. &Grauert, H., Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten.Math. Ann., 140 (1960), 94–123.CrossRefMathSciNetMATHGoogle Scholar
  15. [EH1]
    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. [EH2]
    —, 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. [EH3]
    —, Embeddings for 3-dimensional CR-manifolds, inComplex Analysis and Geometry (Paris, 1997), pp. 223–236. Progr. Math., 188. Birkhäuser, Basel, 2000.Google Scholar
  18. [El]
    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. [Ep1]
    Epstein, C. L., CR-structures on three-dimensional circle bundles.Invent. Math., 109 (1992), 351–403.CrossRefMATHMathSciNetGoogle Scholar
  20. [Ep2]
    —, A relative index for embeddable CR-structures, I; II.Ann. of Math., 147 (1998), 1–59; 61–91.CrossRefMATHMathSciNetGoogle Scholar
  21. [F]
    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.MATHMathSciNetGoogle Scholar
  22. [GH]
    Griffiths, P. &Harris, J.,Principles of Algebraic Geometry. Wiley-Interscience, New York, 1978.MATHGoogle Scholar
  23. [Gr]
    Griffiths, P., The extension problem in complex analysis, II. Embeddings with positive normal bundle.Amer. J. Math., 88 (1966), 366–446.MATHMathSciNetGoogle Scholar
  24. [GR]
    Grauert, H. &Riemenschneider, O., Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen.Invent. Math., 11 (1970), 263–292.CrossRefMathSciNetMATHGoogle Scholar
  25. [Gu]
    Gunning, R. C.,Introduction to Holomorphic Functions of Several Variables, Vol. II. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1990.MATHGoogle Scholar
  26. [Ht]
    Hartshorne, R.,Algebraic Geometry. Graduate Texts in Math., 52. Springer-Verlag, New York-Heidelberg, 1977.MATHGoogle Scholar
  27. [Hv]
    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. [HvL]
    Harvey, F. R. &Lawson, H. B., Jr., On boundaries of complex analytic varieties, I.Ann. of Math., 102 (1975), 223–290.CrossRefMathSciNetGoogle Scholar
  29. [Ka]
    Kato, T.,Perturbation Theory of Linear Operators. Grundlehren Math. Wiss., 132. Springer-Verlag, New York, 1966.Google Scholar
  30. [Kd1]
    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.CrossRefMATHMathSciNetGoogle Scholar
  31. [Kd2]
    —, On stability of compact submanifolds of complex manifolds.Amer. J. Math., 85 (1963), 79–94.MATHMathSciNetGoogle Scholar
  32. [Ki]
    Kiremidjian, G., A direct extension method for CR-structures.Math. Ann., 242 (1979), 1–19.CrossRefMATHMathSciNetGoogle Scholar
  33. [KM]
    Kronheimer, P. B. &Mrowka, T. S., Monopoles and contact structures.Invent. Math., 130 (1997), 209–255.CrossRefMathSciNetMATHGoogle Scholar
  34. [Ko]
    Kohn, J. J., The range of the tangential Cauchy-Riemann operator.Duke Math. J., 53 (1986), 525–545.CrossRefMATHMathSciNetGoogle Scholar
  35. [KR]
    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.CrossRefMathSciNetGoogle Scholar
  36. [Ku]
    Kuranishi, M.,Deformations of Compact Complex Manifolds. Séminaire de Mathématiques Supérieures, 39. Les Presses de l'Université de Montréal, Montreal, 1971.Google Scholar
  37. [Le1]
    Lempert, L., On three-dimensional Cauchy-Riemann manifolds.J. Amer. Math. Soc., 5 (1992), 923–969.CrossRefMATHMathSciNetGoogle Scholar
  38. [Le2]
    —, Embeddings of three-dimensional Cauchy-Riemann manifolds.Math. Ann., 300 (1994), 1–15.CrossRefMATHMathSciNetGoogle Scholar
  39. [Le3]
    —, Algebraic approximations in analytic geometry.Invent. Math., 121 (1995), 335–353.CrossRefMATHMathSciNetGoogle Scholar
  40. [Li]
    Li, H.-L., The stability of embeddings of Cauchy-Riemann manifolds. Thesis, Purdue University, 1995.Google Scholar
  41. [MR1]
    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. [MR2]
    —, Some theorems of algebraicity for complex spaces.J. Math. Soc. Japan, 27 (1975), 167–183.MathSciNetCrossRefMATHGoogle Scholar
  43. [O]
    Ouyang, Y., Ph.D. Thesis, University of Pennsylvania, 1999.Google Scholar
  44. [Pi]
    Pinkham, H. C., Deformations of cones with negative grading.J. Algebra, 30 (1974), 92–102.CrossRefMATHMathSciNetGoogle Scholar
  45. [PS]
    Pardon, W. &Stern, M. A.,L 2-\(\bar \partial \)-cohomology of complex projective varieties.J. Amer. Math. Soc., 4 (1991), 603–621.CrossRefMathSciNetMATHGoogle Scholar
  46. [R]
    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. [Si]
    Siu, Y.-T., Every Stein subvariety admits a Stein neighborhood.Invent. Math., 38 (1976), 89–100.CrossRefMATHMathSciNetGoogle Scholar
  48. [SS]
    Shiffman, B. &Sommese, A. J.,Vanishing Theorems on Complex Manifolds. Progr. Math., 56. Birkhäuser, Boston, Boston, MA, 1985.MATHGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2000

Authors and Affiliations

  • Charles L. Epstein
    • 1
  • Gennadi M. Henkin
    • 2
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Institut de Mathématique de JussieuUniversité de Paris VIParis Cedex 05France

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