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Kobayashi Hyperbolicity

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Nevanlinna Theory in Several Complex Variables and Diophantine Approximation

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 350))

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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.

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Notes

  1. 1.

    Actually, given two points x,y on a complex manifold X there always exists a chain of length one, i.e., a holomorphic map f:Δ(1)→X with x,y∈F(Δ(1)) (see Winkelmann [05]). However, in any case we need to consider for the triangle inequality.

  2. 2.

    This statement was claimed in Babets, V.A., Picard-type theorems for holomorphic mappings, Siberian Math. J. 25 (1984), 195–200. In Theorem 1 the case of f(C)⊂D by the notation there was not dealt with, and it was stated that the quasi-Albanese variety is the product of the Albanese variety and (C ∗)t; this is not true in general.

  3. 3.

    J. Winkelmann, On meromorphic functions which are Brody curves, arXiv:0709.3929.

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Author information

Authors and Affiliations

  1. The University of Tokyo, Tokyo, Japan

    Emeritus Junjiro Noguchi

  2. Tokyo Institute of Technology, Tokyo, Japan

    Emeritus Junjiro Noguchi

  3. Ruhr-University Bochum, Bochum, Germany

    Jörg Winkelmann

Authors
  1. Emeritus Junjiro Noguchi
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  2. Jörg Winkelmann
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Appendix: Program for Mathematica

Appendix: Program for Mathematica

figure a
figure b
figure c
figure d
figure e
figure f
figure g
figure h

An example follows to use the above program to check (7.5.12). Vectors correspond to the exponents of z 5,z 1,z 2,z 3,z 4, in that order.

figure i
figure j

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Noguchi, J., Winkelmann, J. (2014). Kobayashi Hyperbolicity. In: Nevanlinna Theory in Several Complex Variables and Diophantine Approximation. Grundlehren der mathematischen Wissenschaften, vol 350. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54571-2_7

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