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Bergman Kernel on Non-compact Manifolds

Part of the Progress in Mathematics book series (PM, volume 254)

Abstract

We show in Section 6.1 that the asymptotic expansion of the Bergman kernel still holds on compact sets of certain non-compact complete manifolds. In this way we can obtain another proof of some of the holomorphic Morse inequalities. As a corollary, we re-prove the Shiffman-Ji-Bonavero-Takayama criterion for Moishezon manifolds in Section 6.2.

Keywords

Line Bundle Complex Manifold Ahler Manifold Pseudoconvex Domain Bergman Kernel 
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6.5 Bibliographic notes

  1. [2]
    A. Andreotti, Théorèmes de dépendance algébrique sur les espaces complexes pseudo-concaves, Bull. Soc. Math. France 91 (1963), 1–38. Théorème 6MATHMathSciNetGoogle Scholar
  2. [3]
    A. Andreotti and T. Frankel, The Lefschetz theorem on hyperplane sections, Ann. of Math. 69 (1959), 713–717.CrossRefMathSciNetGoogle Scholar
  3. [5]
    A. Andreotti and Y.T. Siu, Projective embeddings of pseudoconcave spaces, Ann. Sc. Norm. Sup. Pisa 24 (1970), 231–278. Theorem 4.1MATHMathSciNetGoogle Scholar
  4. [6]
    A. Andreotti and G. Tomassini, Some remarks on pseudoconcave manifolds, Essays on Topology and Related Topics dedicated to G. de Rham (A. Haefinger and R. Narasimhan, eds.), Springer-Verlag, 1970, pp. 85–104. Theorem 3, p.97Google Scholar
  5. [46]
    R. Bott, On a theorem of Lefschetz, Mich. Math. J. 6 (1959), 211–216.MATHCrossRefMathSciNetGoogle Scholar
  6. [51]
    L. Boutet de Monvel, Intégration des équations de Cauchy-Riemann induites formelles, Séminaire Goulaouic-Lions-Schwartz 1974–1975; Équations aux derivées partielles linéaires et non linéaires, Centre Math., École Polytech., Paris, 1975, pp. Exp. No. 9, 14. p. 5Google Scholar
  7. [53]
    L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Journées: Équations aux Dérivées Partielles de Rennes (1975), Soc. Math. France, Paris, 1976, pp. 123–164. Astérisque, No. 34–35.Google Scholar
  8. [56]
    C. Bănică and O. Stănăşilă, Algebraic methods in the global theory of complex spaces, Wiley, New York, 1976. Prop. VI.2.7Google Scholar
  9. [57]
    D.M. Burns, Global behavior of some tangential Cauchy-Riemann equations, Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977), Lecture Notes in Pure and Appl. Math., vol. 48, Dekker, New York, 1979, pp. 51–56.Google Scholar
  10. [59]
    J. Carlson and P. Griffiths, A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. of Math. 95 (1972), 557–584. § 2CrossRefMathSciNetGoogle Scholar
  11. [67]
    M. Coltoiu, Complete locally pluripolar sets, J. Reine Angew. Math. 412 (1990), 108–112.MATHMathSciNetGoogle Scholar
  12. [68]
    M. Coltoiu and M. TibĂr, Steinness of the universal covering of the complement of a 2-dimensional complex singularity, Math. Ann. 326 (2003), no. 1, 95–104.MATHCrossRefMathSciNetGoogle Scholar
  13. [100]
    P. Eberlein, Lattices in spaces of nonpositive curvature, Ann. of Math. (2) 111 (1980), no. 3, 435–476.CrossRefMathSciNetGoogle Scholar
  14. [101]
    P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109. § 10MATHMathSciNetGoogle Scholar
  15. [104]
    C.L. Epstein and G.M. Henkin, Can a good manifold come to a bad end?, Tr. Mat. Inst. Steklova 235 (2001), no. Anal. i Geom. Vopr. Kompleks. Analiza, 71–93. Theorem 2MathSciNetGoogle Scholar
  16. [110]
    W. Fulton and R. Lazarsfeld, Connectivity and its applications in algebraic geometry, Algebraic geometry (Chicago, Ill., 1980), Lecture Notes in Math., vol. 862, Springer, Berlin, 1981, pp. 26–92.CrossRefGoogle Scholar
  17. [113]
    C. Grant and P. Milman, Metrics for singular analytic spaces, Pacific J. Math. 168 (1995), no. 1, 61–156.MATHMathSciNetGoogle Scholar
  18. [114]
    C. Grant Melles and P. Milman, Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 4, 689–771.MATHMathSciNetGoogle Scholar
  19. [117]
    H. Grauert, Theory of q-convexity and q-concavity, Several Complex Variables VII (H. Grauert, Th. Peternell, and R. Remmert, eds.), Encyclopedia of mathematical sciences, vol. 74, Springer Verlag, 1994. p. 273Google Scholar
  20. [121]
    A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, North-Holland, Amsterdam, 1968.Google Scholar
  21. [123]
    R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Math., vol. 156, Springer-Verlag, Berlin, 1970.MATHGoogle Scholar
  22. [124]
    _____, Algebraic Geometry, Springer-Verlag, Berlin, 1977.MATHGoogle Scholar
  23. [125]
    R. Harvey and B. Lawson, On boundaries of complex varieties I, Ann. of Math.. 102 (1975), 233–290.CrossRefMathSciNetGoogle Scholar
  24. [127]
    D. Heunemann, Extension of the complex structure from Stein manifolds with strictly pseudoconvex boundary, Math. Nachr. 128 (1986), 57–64. Theorem 0.2MATHCrossRefMathSciNetGoogle Scholar
  25. [144]
    J.J. Kohn, The range of the tangential Cauchy-Riemann operators, Duke Math. 53 (1986), 525–545.MATHCrossRefMathSciNetGoogle Scholar
  26. [145]
    J.J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. Math. 81 (1965), 451–472.CrossRefMathSciNetGoogle Scholar
  27. [147]
    J. Kollár, Shafarevich maps and automorphic forms, Princeton Univ. Press, Princeton, NJ, 1995.MATHGoogle Scholar
  28. [149]
    R. Lazarsfeld, Positivity in algebraic geometry. I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 48-49, Springer-Verlag, Berlin, 2004.Google Scholar
  29. [150]
    S. Lefschetz, L’analysis situs et la géométrie algébrique, Dunod, Paris, 1924.MATHGoogle Scholar
  30. [151]
    L. Lempert, On three-dimensional Cauchy-Riemann manifolds, J. Amer. Math. Soc. 5 (1992), no. 4, 923–969.MATHCrossRefMathSciNetGoogle Scholar
  31. [152]
    _____-, Embeddings of three-dimensional Cauchy-Riemann manifolds, Math. Ann. 300 (1994), no. 1, 1–15.MATHCrossRefMathSciNetGoogle Scholar
  32. [153]
    _____-, Algebraic approximations in analytic geometry, Invent. Math. 121 (1995), no. 2, 335–353.MATHCrossRefMathSciNetGoogle Scholar
  33. [161]
    X. Ma and G. Marinescu, Generalized Bergman kernels on symplectic manifolds, C. R. Acad. Sci. Paris 339 (2004), no. 7, 493–498, The full version: math.DG/0411559, Adv. in Math.MATHMathSciNetGoogle Scholar
  34. [172]
    G. Marinescu and T.-C. Dinh, On the compactification of hyperconcave ends and the theorems of Siu-Yau and Nadel, Invent. Math. 164 (2006), 233–248.MATHCrossRefMathSciNetGoogle Scholar
  35. [174]
    G. Marinescu and N. Yeganefar, Embeddability of some strongly pseudoconvex CR manifolds, to appear in Trans. Amer. Math. Soc., Preprint available at arXiv:math.CV/0403044, 2004.Google Scholar
  36. [176]
    J. Milnor, Morse Theory, Ann. Math. Studies, vol. 51, Princeton University Press, 1963.Google Scholar
  37. [178]
    N. Mok, Compactification of complete Kähler surfaces of finite volume satisfying certain curvature conditions, Ann. Math. 129 (1989), 383–425.CrossRefMathSciNetGoogle Scholar
  38. [181]
    A. Nadel and H. Tsuji, Compactification of complete Kähler manifolds of negative Ricci curvature, J. Differential Geom. 28 (1988), no. 3, 503–512. Lemma 2.1MATHMathSciNetGoogle Scholar
  39. [183]
    T. Napier and M. Ramachandran, The L 2-method, weak Lefschetz theorems and the topology of Kähler manifolds, JAMS 11 (1998), no. 2, 375–396.MATHCrossRefMathSciNetGoogle Scholar
  40. [186]
    M.V. Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. 16 (1983), 305–344.MATHMathSciNetGoogle Scholar
  41. [188]
    T. Ohsawa, Holomorphic embedding of compact s.p.c. manifolds into complex manifolds as real hypersurfaces, Differential geometry of submanifolds (Kyoto, 1984), Lecture Notes in Math., vol. 1090, Springer, Berlin, 1984, pp. 64–76.CrossRefGoogle Scholar
  42. [207]
    H. Rossi, Attaching analytic spaces to an analytic space along a pseudoconcave boundary, Proc. Conf. Complex. Manifolds (Minneapolis), Springer-Verlag, New York, 1965, pp. 242–256. Th. 3, p. 245Google Scholar
  43. [228]
    Y.T. Siu and S.T. Yau, Compactification of negatively curved complete Kähler manifolds of finite volume, Ann. Math. Stud., vol. 102, Princeton Univ. Press, 1982, pp. 363–380.MathSciNetGoogle Scholar
  44. [244]
    R. Todor, I. Chiose, and G. Marinescu, Morse inequalities for covering manifolds, Nagoya Math. J. 163 (2001), 145–165.MATHMathSciNetGoogle Scholar
  45. [255]
    H. Wu, Negatively curved Kähler manifolds, Notices. Amer. Math. Soc. 14 (1967), 515.Google Scholar

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