Journal of the Mathematical Society of Japan

Pseudo Kobayashi hyperbolicity of subvarieties of general type on abelian varieties

Katsutoshi YAMANOI

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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.


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

pseudo Kobayashi hyperbolicity tautness Nevanlinna theory


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.

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