Journal of the Mathematical Society of Japan

Pseudo Kobayashi hyperbolicity of subvarieties of general type on abelian varieties

Katsutoshi YAMANOI

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Abstract

We prove that the Kobayashi pseudo distance of a closed subvariety $X$ of an abelian variety $A$ is a true distance outside the special set $\operatorname{Sp}(X)$ of $X$, where $\operatorname{Sp}(X)$ is the union of all positive dimensional translated abelian subvarieties of $A$ which are contained in $X$. More strongly, we prove that a closed subvariety $X$ of an abelian variety is taut modulo $\operatorname{Sp}(X)$; Every sequence $f_n:{\mathbb{D}}\to X$ of holomorphic mappings from the unit disc ${\mathbb{D}}$ admits a subsequence which converges locally uniformly, unless the image $f_n(K)$ of a fixed compact set $K$ of ${\mathbb{D}}$ eventually gets arbitrarily close to $\operatorname{Sp}(X)$ as $n$ gets larger. These generalize a classical theorem on algebraic degeneracy of entire curves in irregular varieties.

Note

The author was supported by JSPS Grant-in-Aid for Scientific Research (C), 24540069 and by JSPS Grant-in-Aid for Scientific Research (B), 17H02842.

Article information

Source
J. Math. Soc. Japan, Volume 71, Number 1 (2019), 259-298.

Dates
Received: 10 August 2016
Revised: 17 August 2017
First available in Project Euclid: 20 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1542704620

Digital Object Identifier
doi:10.2969/jmsj/75817581

Mathematical Reviews number (MathSciNet)
MR3909921

Zentralblatt MATH identifier
07056564

Subjects
Primary: 32Q45: Hyperbolic and Kobayashi hyperbolic manifolds
Secondary: 32H30: Value distribution theory in higher dimensions {For function- theoretic properties, see 32A22} 14K12: Subvarieties

Keywords
pseudo Kobayashi hyperbolicity tautness Nevanlinna theory

Citation

YAMANOI, Katsutoshi. Pseudo Kobayashi hyperbolicity of subvarieties of general type on abelian varieties. J. Math. Soc. Japan 71 (2019), no. 1, 259--298. doi:10.2969/jmsj/75817581. https://projecteuclid.org/euclid.jmsj/1542704620


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