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Kobayashi Hyperbolicity

  • Junjiro Noguchi
  • Jörg Winkelmann
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 350)

Abstract

We describe a way to construct Kobayashi hyperbolic algebraic varieties, and one to construct Kobayashi hyperbolic projective hypersurfaces. Here we will effectively use the theory of entire curves given in the previous chapters. The results of this chapter can be proved so far now only through Nevanlinna theory of entire curves.

Keywords

Meromorphic Function Complex Manifold Compact Riemann Surface Compact Complex Manifold Algebraic Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

Ax, J.

  1. [72]
    Some topics in differential algebraic geometry II, Am. J. Math. 94 (1972), 1205–1213. MathSciNetCrossRefMATHGoogle Scholar

Azukawa, K. and Suzuki, Masaaki

  1. [80]
    Some examples of algebraic degeneracy and hyperbolic manifolds, Rocky Mt. J. Math. 10 (1980), 655–659. MathSciNetCrossRefMATHGoogle Scholar

Barth, T.

  1. [72]
    The Kobayashi distance induces the standard topology, Proc. Am. Math. Soc. 35 (1972), 439–441. MathSciNetCrossRefMATHGoogle Scholar

Bloch, A.

  1. [26a]
    Sur les système de fonctions holomorphes à variétés linéaires lacunaires, Ann. Sci. Éc. Norm. Super. 43 (1926a), 309–362. MathSciNetMATHGoogle Scholar

Brody, R.

  1. [78]
    Compact manifolds and hyperbolicity, Trans. Am. Math. Soc. 235 (1978), 213–219. MathSciNetMATHGoogle Scholar

Brody, R. and Green, M.

  1. [77]
    A family of smooth hyperbolic hypersurfaces in P 3, Duke Math. J. 44 (1977), 873–874. MathSciNetCrossRefMATHGoogle Scholar

Cartan, H.

  1. [28]
    Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires, Ann. Sci. Éc. Norm. Super. 45 (1928), 255–346. MathSciNetMATHGoogle Scholar

Clunie, J. and Hayman, W.K.

  1. [66]
    The spherical derivative of integral and meromorphic functions, Comment. Math. Helv. 40 (1966), 117–148. MathSciNetCrossRefMATHGoogle Scholar

Demailly, J.P. and El Goul, J.

  1. [00]
    Hyperbolicity of generic surfaces of high degree in projective 3-space, Am. J. Math. 122 (2000), 515–546. CrossRefMATHGoogle Scholar

Dufresnoy, M.J.

  1. [44]
    Théorie nouvelle des familles complexes normales: Applications à l’étude des fonctions algébroïdes, Ann. Sci. Éc. Norm. Super. 61 (1944), 1–44. MathSciNetMATHGoogle Scholar

Duval, J.

  1. [04]
    Une sextique hyperbolique dans P 3(C), Math. Ann. 330 no. 3 (2004), 473–476. MathSciNetCrossRefMATHGoogle Scholar
  2. [08]
    Sur le lemme de Brody, Invent. Math. 173 (2008), 305–314. MathSciNetCrossRefMATHGoogle Scholar

Eremenko, A.E.

  1. [99]
    A Picard type theorem for holomorphic curves, Period. Math. Hung. 38 (1999), 39–42. MathSciNetCrossRefMATHGoogle Scholar

Fujimoto, H.

  1. [72a]
    On holomorphic maps into a taut complex space, Nagoya Math. J. 46 (1972a), 49–61. MathSciNetGoogle Scholar
  2. [01]
    A family of hyperbolic hypersurfaces in the complex projective space, Complex Var. Elliptic Equ. 43 (2001), 273–283. MathSciNetCrossRefMATHGoogle Scholar

Green, M.L.

  1. [77]
    The hyperbolicity of the complement of 2n+1 hyperplanes in general position in P n, and related results, Proc. Am. Math. Soc. 66 (1977), 109–113. MATHGoogle Scholar
  2. [78]
    Holomorphic maps to complex tori, Am. J. Math. 100 (1978), 615–620. CrossRefMATHGoogle Scholar

Kobayashi, S.

  1. [70]
    Hyperbolic Manifolds and Holomorphic Mappings, Marcel Dekker, New York, 1970. MATHGoogle Scholar
  2. [98]
    Hyperbolic Complex Spaces, Grundl. Math. Wissen. 318, Springer, Berlin, 1998. CrossRefMATHGoogle Scholar

Kodama, A.

  1. [83]
    Remarks on homogeneous hyperbolic complex manifolds, Tohoku Math. J. 35 (1983), 181–186. MathSciNetCrossRefMATHGoogle Scholar

Lang, S.

  1. [87]
    Introduction to Complex Hyperbolic Spaces, Springer, Berlin, 1987. CrossRefMATHGoogle Scholar

McQuillan, M.

  1. [98]
    Diophantine approximations and foliations, Publ. Math. IHÉS 87 (1998), 121–174. MathSciNetCrossRefMATHGoogle Scholar

Nadel, A.

  1. [89b]
    Hyperbolic surfaces in \({\bf P}^{3}\), Duke Math. J. 58 (1989b), 749–771. MathSciNetCrossRefMATHGoogle Scholar

Noguchi, J.

  1. [88]
    Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces, Invent. Math. 93 (1988), 15–34. MathSciNetCrossRefMATHGoogle Scholar
  2. [96]
    On Nevanlinna’s second main theorem, Geometric Complex Analysis, Proc. 3rd MSJ-IRI Hayama, 1995, J. Noguchi et al. (Eds.), pp. 489–503, World Scientific, Singapore, 1996. Google Scholar

Noguchi, J. and Ochiai, T.

  1. [90]
    Geometric Function Theory in Several Complex Variables, Transl. Math. Mono. 80, Am. Math. Soc., Providence, 1990. MATHGoogle Scholar

Ochiai, T.

  1. [77]
    On holomorphic curves in algebraic varieties with ample irregularity, Invent. Math. 43 (1977), 83–96. MathSciNetCrossRefMATHGoogle Scholar

Ochiai, T. and Noguchi, J.

  1. [84]
    Geometric Function Theory in Several Variables (in Japanese), Iwanami Shoten, Tokyo, 1984. Google Scholar

Royden, H.L.

  1. [71]
    Remarks on the Kobayashi metric, Several Complex Variables II Maryland 1970, Lecture Notes in Math. 185, pp. 125–137, Springer, Berlin, 1971. CrossRefGoogle Scholar

Shiffman, B. and Zaidenberg, M.

  1. [02a]
    Hyperbolic hypersurfaces in P n of Fermat-Waring type, Proc. Am. Math. Soc. 130 (2002a), 2031–2035. MathSciNetCrossRefMATHGoogle Scholar
  2. [02b]
    Constructing low degree hyperbolic surfaces in P 3, Special issue for S.-S. Chern, Houst. J. Math. 28 (2002b), 377–388. MathSciNetMATHGoogle Scholar

Shirosaki, M.

  1. [98]
    On some hypersurfaces and holomorphic mappings, Kodai Math. J. 21 (1998), 29–34. MathSciNetCrossRefMATHGoogle Scholar

Siu, Y.-T.

  1. [76]
    Every Stein subvariety has a Stein neighborhood, Invent. Math. 38 (1976), 89–100. MathSciNetCrossRefMATHGoogle Scholar

Voisin, C.

  1. [96]
    On a conjecture of Clemens on rational curves on hypersurfaces, J. Differ. Geom. 44 (1996), 200–213; A correction “On a conjecture of Clemens on rational curves on hypersurfaces”, J. Diff. Geom. 49 (1998), 601–611. MathSciNetMATHGoogle Scholar

Winkelmann, J.

  1. [90]
    The Kobayashi-pseudodistance on homogeneous manifolds, Manuscr. Math. 68 (1990), 117–134. MathSciNetCrossRefMATHGoogle Scholar
  2. [05]
    Non-degenerate maps and sets, Math. Z. 249 (2005), 783–795. MathSciNetCrossRefMATHGoogle Scholar
  3. [07]
    On Brody and entire curves, Bull. Soc. Math. Fr. 135 (2007), 25–46. MathSciNetMATHGoogle Scholar

Yosida, K.

  1. [34]
    On a class of meromorphic functions, Proc. Phys. Math. Soc. Jpn. 16 (1934), 227–235. Google Scholar

Zaidenberg, M.G.

  1. [89]
    Stability of hyperbolic imbeddedness and construction of examples, Math. USSR Sb. 63 (1989), 351–361. MathSciNetCrossRefGoogle Scholar

Zalcman, L.

  1. [98]
    Normal families: New perspectives, Bull. Am. Math. Soc. 35 no. 3 (1998), 215–230. MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Junjiro Noguchi
    • 1
    • 2
  • Jörg Winkelmann
    • 3
  1. 1.The University of TokyoTokyoJapan
  2. 2.Tokyo Institute of TechnologyTokyoJapan
  3. 3.Ruhr-University BochumBochumGermany

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