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On holomorphic curves in semi-Abelian varieties

On holomorphic curves in semi-Abelian varieties Math. Z. 228, 713–721 (1998) c Springer-Verlag 1998 Junjiro Noguchi Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan (e-mail: noguchi@ms.u-tokyo.ac.jp) Received 1 October 1996; in final form 3 February 1997 Introduction The purpose of this paper is to prove Main Theorem. Let D be a non-zero algebraic effective reduced divisor of a semi-Abelian variety A over the complex number field C. Let f : C ! AnD be an arbitrary holomorphic mapping. (i) The Zariski closure X (f) of the image of f in A is a translate of a proper semi-Abelian subvariety of A, and X (f) \ D = ;. (ii) In special, if D has a non-empty intersection with any translate of any positive dimensional semi-Abelian subvariety of A, then f : C ! A nD is constant. In the course of the proof we need a generalization of theta functions for semi-Abelian varieties (see Lemma (2.1) and Remark after it). Moreover, as an application of the proof of the Main Theorem, we prove a structure theorem for the locus (sometimes, called the exceptional set) of A n D which contains the images of all possible non-constant entire holomorphic curves in A http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

On holomorphic curves in semi-Abelian varieties

Mathematische Zeitschrift , Volume 228 (4) – Aug 1, 1998

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References (18)

Publisher
Springer Journals
Copyright
Copyright © 1998 by Springer-Verlag Berlin Heidelberg
Subject
Legacy
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/PL00004640
Publisher site
See Article on Publisher Site

Abstract

Math. Z. 228, 713–721 (1998) c Springer-Verlag 1998 Junjiro Noguchi Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan (e-mail: noguchi@ms.u-tokyo.ac.jp) Received 1 October 1996; in final form 3 February 1997 Introduction The purpose of this paper is to prove Main Theorem. Let D be a non-zero algebraic effective reduced divisor of a semi-Abelian variety A over the complex number field C. Let f : C ! AnD be an arbitrary holomorphic mapping. (i) The Zariski closure X (f) of the image of f in A is a translate of a proper semi-Abelian subvariety of A, and X (f) \ D = ;. (ii) In special, if D has a non-empty intersection with any translate of any positive dimensional semi-Abelian subvariety of A, then f : C ! A nD is constant. In the course of the proof we need a generalization of theta functions for semi-Abelian varieties (see Lemma (2.1) and Remark after it). Moreover, as an application of the proof of the Main Theorem, we prove a structure theorem for the locus (sometimes, called the exceptional set) of A n D which contains the images of all possible non-constant entire holomorphic curves in A

Journal

Mathematische ZeitschriftSpringer Journals

Published: Aug 1, 1998

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