Two Applications of Algebraic Geometry to Entire Holomorphic Mappings

Conference paper


In this paper we shall prove two theorems concerning holomorphic mappings of large open sets of ℂk into algebraic varieties. Both are in response to well-known outstanding problems, and we feel that the techniques introduced should in each case have further applications.


Holomorphic Mapping Algebraic Variety Elliptic Curf Abelian Variety Holomorphic Sectional Curvature 
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  1. [1]
    A. Bloch, Sur les systémes de fonctions uniformes satisfaisant â l’equation d’une variété algébrique dont l’irrégularité dépasse la dimension. J. de Math. V, 19–66 (1926).Google Scholar
  2. [2]
    F. Bogomolov, Families of curves on a surface of general type. Sov. Math. Dokl. 18, 1294–1297 (1977).zbMATHGoogle Scholar
  3. [3]
    R. Brody, Intrinsic metrics and measures on compact complex manifolds. Ph.D. Thesis, Harvard Univ., 1975.Google Scholar
  4. [4]
    G. Castelnuovo and F. Enriques, Grundeigenschaften der Algebraischen Flächen. In Encyklop. d. Math. Wissensch., Vol. 3, 1903, pp. 635–768.Google Scholar
  5. [5]
    M. Cowen, Families of negatively curved Hermitian manifolds. Proc. Amer. Math. Soc. 39 362–366 (1973).MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    M. Cowen and P. Griffiths, Holomorphic curves and metrics of negative curvature. J. Analyse Math.Google Scholar
  7. [7]
    P. Deligne and D. Mumford, The irreducibility of the space of curves of a given genus. Publ. Math. IHES 36, 75–109 (1969).MathSciNetzbMATHGoogle Scholar
  8. [8]
    I. Dolgacev, Weighted Projective Varieties, to appear.Google Scholar
  9. [9]
    R. Donagi, On the geometry of Grassmannians. Duke Math. J. 44, 795–837 (1977).MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    H. Grauert and H. Reckziegel, Hermitesche Metriken and normale familien holomorpher Abbildungen. Math. Z. 89, 108–125 (1965).MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    M. Green, Holomorphic maps into complex projective space omitting hyperplanes. Trans. Amer. Math. Soc. 169, 89–103 (1972).MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    M. Green, Holomorphic maps into complex tori. Amer. J. Math. 100, 615–620 (1978).MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    P. Griffiths, Complex differential and integral geometry and curvature integrals associated to singularities of complex analytic varieties. Duke Math. J. 45, 427–512 (1978).MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    P. Griffiths, Holomorphic mappings into canonical algebraic varieties. Ann. of Math. (2) 93, 439–458 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    P. Griffiths and J. Harris, Local differential geometry and algebraic geometry. Ann. Ec. Norm sup. (Dec. 1979).Google Scholar
  16. [16]
    P. Griffiths and J. Harris, On the variety of special linear systems on a general algebraic curve. Duke Math. J. to appear.Google Scholar
  17. [17]
    P. Griffiths and J. Harris, Principles of Algebraic Geometry. Wiley, New York, 1978.zbMATHGoogle Scholar
  18. [18]
    S. Iitaka, On D-dimensions of algebraic varieties. J. Math. Soc. Japan 23, 356–373 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    S. Kleiman, Rigorous foundations of Schubert’s enumerative calculus, in Mathematical developments arising from Hilbert’s problems. In Proc. of Symposia, AMS, Vol. 27.Google Scholar
  20. [20]
    S. Kobayashi, Intrinsic distances, measures and geometric function theory. Bull. Amer. Math. Soc. 82, 357–416 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    S. Kobayashi and T. Ochiai, Mappings into compact complex manifolds with negative first Chem class. J. Math. Soc. Japan 23, 137–148 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    A. Mayer, Families of K-3 surfaces.Google Scholar
  23. [23]
    S. Mori, Projective manifolds with ample tangent bundles. To appear.Google Scholar
  24. [24]
    J. Noguchi, Meromorphic mappings into a compact complex space. Hiroshima Math. J. 7 (2), 411–425 (1977).MathSciNetzbMATHGoogle Scholar
  25. [25]
    T. Ochiai, On holomorphic curves in algebraic varieties with ample irregularity. Invent. Math. 43, S3–96 (1977).MathSciNetCrossRefGoogle Scholar
  26. [26]
    E. Picard, Sur une propriété des fonctions uniformes, liées par une relation algébrique. Compt. Rend. 91, 724–726 (1880).Google Scholar
  27. [27]
    B. Saint-Donat, Projective models of K-3 surfaces.Google Scholar
  28. [28]
    F. Sakai, Symmetry powers of the cotangent bundle and classification of algebraic varieties. In Proc. Copenhagen Summer Meeting in Algebraic Geometry, 1978.Google Scholar
  29. [29]
    H. Wu, A remark on holomorphic sectional curvature. Indiana Math. J. 22, 1103–1108 (1972/73).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1980

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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