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Monge-Ampère Operators, Lelong Numbers and Intersection Theory

  • Jean-Pierre Demailly
Part of the The University Series in Mathematics book series (USMA)

Abstract

This chapter is a survey article on the theory of Lelong numbers, viewed as a tool for studying intersection theory by complex differential geometry. We have not attempted to make an exhaustive compilation of the existing literature on the subject, nor to present a complete account of the state-of-the-art. Instead, we have tried to present a coherent unifying frame for the most basic results of the theory, based in part on our earlier works [7–10] and on Siu’s fundamental work [30]. To a large extent, the asserted results are given with complete proofs, many of them substantially shorter and simpler than their original counterparts. We only assume that the reader has some familiarity with differential calculus on complex manifolds and with the elementary facts concerning analytic sets and plurisubharmonic functions. The reader can consult Lelong’s books [25, 26] for an introduction to the subject. Most of our results still work on arbitrary complex analytic spaces, provided that suitable definitions are given for currents, plurisubharmonic functions, etc., in this more general situation. We have refrained ourselves from doing so for simplicity of exposition; we refer the reader to Ref. 9 for the technical definitions required in the context of analytic spaces.

Keywords

Complex Manifold Comparison Theorem Plurisubharmonic Function Positive Current Finite Mass 
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References

  1. 1.
    A. Azhari, Sur la conjecture de Chudnovsky-Demailly et les singularités des hypersurfaces algébriques, Ann. Inst. Fourier 40, 106–117 (1990).MathSciNetCrossRefGoogle Scholar
  2. 2.
    E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math. 37, 1–44 (1976).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149, 1–41 (1982).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    E. Bombieri, Algebraic values of meromorphic maps, Invent. Math. 10, 267–287 (1970); Addendum, Invent. Math. 11, 163–166 (1970).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    S. S. Chern, H. I. Levine, and L. Nirenberg, Intrinsic norms on a complex manifold, Global Analysis (papers in honor of K. Kodaira), pp. 119–139, Univ. of Tokyo Press, Tokyo, 1969.Google Scholar
  6. 6.
    G. V. Chudnovsky, Singular points on complex hypersurfaces and multidimensional Schwarz lemma, Sérn. Delange-Pisot-Poitou, 21e année, 1979/80, Progress in Math., No. 12, pp. 29–69 (Marie-José Bertin ed.), Birkhäuser (1981).Google Scholar
  7. 7.
    J.-P. Demailly, Formules de Jensen en plusieurs variables et applications arithmétiques, Bull. Soc. Math. France 110, 75–102 (1982).MathSciNetMATHGoogle Scholar
  8. 8.
    J.-P. Demailly, Sur les nombres de Lelong associés à l’image directe d’un courant positif fermé, Ann. Inst. Fourier (Grenoble) 32, 37–66 (1982).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    J.-P. Demailly, Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines, Mém. Soc. Math. France (N.S.) 19, 1–124 (1985).Google Scholar
  10. 10.
    J.-P. Demailly, Nombres de Lelong généralisés, théorèmes d’intégralité et d’analyticité, Acta Math. 159, 153–169 (1987).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    J.-P. Demailly, Singular Hermitian Metrics on Positive Line Bundles, Lecture Notes in Math. Vol. 1507, Springer-Verlag, Berlin and New York (1992).Google Scholar
  12. 12.
    J.-P. Demailly, A numerical criterion for very ample line bundles, preprint No. 153, Institut Fourier, Univ. Grenoble I (1990) to appear in J. Diff. Geom. (1992).Google Scholar
  13. 13.
    J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Alg. Geom. 1, 361–409 (1992).MathSciNetMATHGoogle Scholar
  14. 14.
    R. N. Draper, Intersection theory in analytic geometry, Math. Ann. 180, 175–203 (1969).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    H. El Mir, Sur le prolongement des courants positifs fermés, Acta Math. 153, 1–45 (1984).MathSciNetMATHCrossRefGoogle Scholar
  16. 15a.
    H. Esnault and E. Viehweg, Sur une minoration du degré d’hypersurfaces s’annulant en certains points, Math. Ann. 263, 75–86 (1983).MathSciNetMATHCrossRefGoogle Scholar
  17. 16.
    H. Federer. Geometric Measure Theory, Springer-Verlag, Berlin and New York (1969).MATHGoogle Scholar
  18. 17.
    R. Harvey, Holomorphic chains and their boundaries, Proc. Symp. Pure Math., Vol. 30, Part 1, pp. 309–382, Several Complex Variables (R. O. Wells, ed.), American Mathematical Society, Providence, RI (1977).Google Scholar
  19. 18.
    L. Hörmander, An introduction to Complex Analysis in several variables, 3rd ed., North-Holland, Amsterdam (1990).MATHGoogle Scholar
  20. 19.
    J. R. King, A residue formula for complex subvarieties, in Proc. Carolina Conf. on Holomorphic Mappings and Minimal Surfaces, pp. 43–56, University of North Carolina, Chapel Hill (1970).Google Scholar
  21. 20.
    C. O. Kiselman, The partial Legendre transformation for plurisubharmonic functions, Invent. Math. 39, 137–148 (1978).MathSciNetCrossRefGoogle Scholar
  22. 21.
    C. O. Kiselman, Densité des fonctions plurisousharmoniques, Bull. Soc. Math. France 107, 295–304 (1979).MathSciNetMATHGoogle Scholar
  23. 22.
    C. O. Kiselman, Sur la Définition de l’Opérateur de Monge-Ampère Complexe, Lecture Notes in Math., Vol. 1094, pp. 139–150, Springer-Verlag, Berlin (1984).Google Scholar
  24. 23.
    C. O. Kiselman, Un nombre de Lelong raffiné, preprint Uppsala 1986, Sém. d’Analyse Complexe et Géométrie 1985–87, Fac. des Sciences de Tunis et Fac. des Sciences et Techniques de Monastir, Maroc (1987).Google Scholar
  25. 24.
    P. Lelong, Intégration sur un ensemble analytique complexe, Bull. Soc. Math. France 85, 239–262 (1957).MathSciNetMATHGoogle Scholar
  26. 25.
    P. Lelong, Fonctionnelles analytiques et fonctions entières (n variables), in Sém. Math. Supérieures, 6e session, Presses Univ. Montreal (1968).Google Scholar
  27. 26.
    P. Lelong, Plurisubharmonic Functions and Positive Differential Forms, Gordon and Breach, New York and Dunod, Paris (1969).MATHGoogle Scholar
  28. 27.
    R. Remmert, Projectionen analytischer Mengen, Math. Ann. 130, 410–441 (1956).MathSciNetMATHCrossRefGoogle Scholar
  29. 28.
    R. Remmert, Holomorphe und meromorphe Abbildungen komplexer Räume, Math. Ann. 133, 328–370 (1957).MathSciNetMATHCrossRefGoogle Scholar
  30. 29.
    N. Sibony, Quelques problèmes de prolongement de courants en analyse complexe, Duke Math. J. 52, 157–197 (1985).MathSciNetMATHCrossRefGoogle Scholar
  31. 30.
    Y. T. Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27, 53–156 (1974).MathSciNetMATHCrossRefGoogle Scholar
  32. 31.
    H. Skoda, Sous-ensembles analytiques d’ordre fini ou infini dans n , Bull. Soc. Math. France 100, 353–408 (1972).MathSciNetMATHGoogle Scholar
  33. 32.
    H. Skoda, Estimations L 2 pour l’opérateur ô et applications arithmétiques, in Sém. P. Lelong (Analyse), Lecture Notes in Math., Vol. 538, pp. 314–323, Springer-Verlag, Berlin (1977).Google Scholar
  34. 33.
    H. Skoda, Prolongement des courants positifs fermés de masse finie, Invent. Math. 66, 361–376 (1982).MathSciNetMATHCrossRefGoogle Scholar
  35. 34.
    W. Stoll, The multiplicity of a holomorphic map, Invent. Math. 2, 15–58 (1966).MathSciNetMATHCrossRefGoogle Scholar
  36. 35.
    P. Thie. The Lelong number of a point of a complex analytic set, Math. Ann.112, 269–312 (1967).MathSciNetCrossRefGoogle Scholar
  37. 36.
    M. Waldschmidt, Propriétés arithmétiques des fonctions de plusieurs variables (II), in Sém. P. Lelong (Analyse), Lecture Notes in Math., Vol. 538, pp. 108–135, Springer-Verlag, Berlin (1977).Google Scholar
  38. 37.
    M. Waldschmidt, Nombres transcendants et groupes algébriques, Astérisque No. 69–70 (1970).Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Jean-Pierre Demailly
    • 1
  1. 1.Institut FourierUniversité de Grenoble ISaint-Martin d’HèresFrance

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