Uniqueness of holomorphic curves into abelian varieties
Authors:
Matthew Dulock and Min Ru
Journal:
Trans. Amer. Math. Soc. 363 (2011), 131142
MSC (2000):
Primary 32H30; Secondary 14K20
Published electronically:
August 24, 2010
MathSciNet review:
2719675
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Abstract: In this paper, we first give a slight improvement of Yamanoi's truncated second main theorem for holomorphic maps into abelian varieties. We then use the result to study the uniqueness problem for such maps. The results obtained generalize and improve E. M. Schmid's uniqueness theorem for holomorphic maps into elliptic curves. In the last section, we consider algebraic dependence for a finite collection of holomorphic curves into an abelian variety.
 [Ai1]
Yoshihiro
Aihara, A unicity theorem for meromorphic mappings into
compactified locally symmetric spaces, Kodai Math. J.
14 (1991), no. 3, 392–405. MR 1131922
(92j:32097), 10.2996/kmj/1138039463
 [Ai2]
Yoshihiro
Aihara, Algebraic dependence of meromorphic mappings in value
distribution theory, Nagoya Math. J. 169 (2003),
145–178. MR 1962526
(2004g:32015)
 [BiL]
Christina
Birkenhake and Herbert
Lange, Complex abelian varieties, 2nd ed., Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], vol. 302, SpringerVerlag, Berlin, 2004. MR 2062673
(2005c:14001)
 [Dr]
S.
J. Drouilhet, A unicity theorem for meromorphic
mappings between algebraic varieties, Trans.
Amer. Math. Soc. 265 (1981), no. 2, 349–358. MR 610953
(82m:32021), 10.1090/S00029947198106109537
 [Fu]
Hirotaka
Fujimoto, Uniqueness problem with truncated multiplicities in value
distribution theory, Nagoya Math. J. 152 (1998),
131–152. MR 1659377
(99m:32029)
 [GrH]
Phillip
Griffiths and Joseph
Harris, Principles of algebraic geometry, Wiley Classics
Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978
original. MR
1288523 (95d:14001)
 [KoO]
Shoshichi
Kobayashi and Takushiro
Ochiai, Mappings into compact manifolds with negative first Chern
class, J. Math. Soc. Japan 23 (1971), 137–148.
MR
0288316 (44 #5514)
 [NoWY]
Junjiro
Noguchi, Jörg
Winkelmann, and Katsutoshi
Yamanoi, The second main theorem for holomorphic curves into
semiabelian varieties. II, Forum Math. 20 (2008),
no. 3, 469–503. MR 2418202
(2009f:32031), 10.1515/FORUM.2008.024
 [Ohb]
Akira
Ohbuchi, Some remarks on ample line bundles on abelian
varieties, Manuscripta Math. 57 (1987), no. 2,
225–238. MR
871633 (87m:14051), 10.1007/BF02218082
 [Sc]
Edwardine
Michele Schmid, Some theorems on value distributions of meromorphic
functions, Math. Z. 120 (1971), 61–92. MR 0284583
(44 #1808)
 [SiY]
YumTong
Siu and SaiKee
Yeung, A generalized Bloch’s theorem and the hyperbolicity of
the complement of an ample divisor in an abelian variety, Math. Ann.
306 (1996), no. 4, 743–758. MR 1418351
(97g:32028), 10.1007/BF01445275
 [Ya]
Katsutoshi
Yamanoi, Holomorphic curves in abelian varieties and intersections
with higher codimensional subvarieties, Forum Math.
16 (2004), no. 5, 749–788. MR 2096686
(2005j:32017), 10.1515/form.2004.035
 [Ai1]
 Aihara, Y. A Unicity Theorem for Meromorphic Mappings into Compactified Locally Symmetric Spaces. Kodai Math J. 14 (1991) no. 3, 392405. MR 1131922 (92j:32097)
 [Ai2]
 Aihara, Y. Algebraic Dependence of Meromorphic Mappings in Value Distribution Theory. Nagoya Math. J. 169 (2003) 145178. MR 1962526 (2004g:32015)
 [BiL]
 Birkenhake, C. and Lange, H. Complex Abelian Varieties. Springer (2003). MR 2062673 (2005c:14001)
 [Dr]
 Drouilhet, S.J. A Unicity Theorem for Meromorphic Maps Between Algebraic Varieties. Trans. Amer. Math Soc. 265 (1981) no. 2, 349358. MR 610953 (82m:32021)
 [Fu]
 Fujimoto, H. Uniqueness Problem with Truncated Multiplicities in Value Distribution Theory. Nagoya Math J. 152 (1998) 131152. MR 1659377 (99m:32029)
 [GrH]
 Griffiths, Ph. and Harris, J. Principles of Algebraic Geometry. John Wiley and Sons (1997). MR 1288523 (95d:14001)
 [KoO]
 Kobayashi, S. and Ochiai, T. Mappings into Compact Complex Manifolds with Negative First Chern Class. J. Math Soc. Japan 23 (1971) 137148. MR 0288316 (44:5514)
 [NoWY]
 Noguchi, J., Winklemann, J. and Yamanoi, K. The Second Main Theorem for Holomorphic Curves into SemiAbelian Varieties II. Forum Math. 20 (2008) no. 3, 469503. MR 2418202 (2009f:32031)
 [Ohb]
 Ohbuchi, A. Some Remarks on Ample Line Bundles of Abelian Varieties. Manuscripta Math. 57 (1987) no. 2, 225238. MR 871633 (87m:14051)
 [Sc]
 Schmid, E.M. Some Theorems of Value Distributions of Meromorphic Functions. Math Z. 120 (1971) 6192. MR 0284583 (44:1808)
 [SiY]
 Siu, Y.T.and Yeung, S.K. Generalized Bloch's Theorem and the Hyperbolicity of the Complement of an Ample Divisor in an Abelian Variety. Math Ann. 306 (1996) no. 4, 743758. MR 1418351 (97g:32028)
 [Ya]
 Yamanoi, K. Holomorphic Curves in Abelian Varieties and Intersections with Higher Codimensional Subvarieties. Forum Math 16 (2004) no. 5, 749788. MR 2096686 (2005j:32017)
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Additional Information
Matthew Dulock
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204
Address at time of publication:
Department of Mathematics, University of North Texas, Denton, Texas 762035017
Email:
dulock11@math.uh.edu
Min Ru
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204
Email:
minru@math.uh.edu
DOI:
http://dx.doi.org/10.1090/S000299472010048883
Keywords:
Abelian variety,
uniqueness theorem,
holomorphic mappings,
algebraic dependency,
Nevanlinna theory
Received by editor(s):
July 3, 2007
Received by editor(s) in revised form:
July 14, 2008
Published electronically:
August 24, 2010
Additional Notes:
The second author was supported in part by NSA under grant number H982300910004.
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
