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A generalized Bloch’s theorem and the hyperbolicity of the complement of an ample divisor in an abelian variety. (English) Zbl 0882.32009
Math. Ann. 306, No. 4, 743-758 (1996); erratum ibid. 326, No. 1, 205-207 (2003).
The purpose of this paper is to prove the following generalization of Bloch’s theorem: If the image of a holomorphic map \(f\) from \(\mathbb{C}\) to an abelian variety is Zariski dense, then the image of the differential of any order of the map \(f\) is invariant under any translation of the abelian variety.
As a corollary the authors prove the next statement: The image from \(\mathbb{C}\) to an abelian variety with Zariski dense image must intersect any hypersurface of the abelian variety.
This theorem together with Bloch’s theorem implies Lang’s conjecture [S. Lang: ‘Number theory’, III: Diophantine geometry. (1991; Zbl 0744.14012)].

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
14K25 Theta functions and abelian varieties
Full Text: DOI EuDML
[1] [B26] A. Bloch: Sur les systèmes de fonctions uniformes satisfaisant à l’équation d’une variété algébrique dont l’irrégularité dépasse la dimension. J. de Math.5 (1926), 19-66
[2] [F91] G. Faltings: Diophantine approximations on abelian varieties, Ann. of Math.129 (1991), 549-576 · Zbl 0734.14007
[3] [Gr78] M. Green: Holomorphic maps to complex tori. Amer. J. Math.100 (1978) 615-620 · Zbl 0384.32007
[4] [GG79] M. Green, P. Griffiths: Two applications of algebraic geometry to entire holomorphic mappings. The Chern Symposium 1979, Proc. Internat. Sympos., Berkeley, 1979, Springer-Verlag 1980
[5] [K80] Y. Kawamata: On Bloch’s conjecture. Invent. Math.57 (1980), 97-100 · Zbl 0569.32012
[6] [La72] S. Lang: Introduction to Algbraic and Abelian Functions, Addison-Wesley, 1972
[7] [La91] S. Lang: Number Theory III. Encyclop. Math. Sc. vol. 60 (1991) Springer-Verlag
[8] [McQ93] M. McQuillan: A new proof of the Bloch conjecture, J. Alg. Geom.5 (1996), 107-117 · Zbl 0862.14027
[9] [NO90] J. Noguchi, T. Ochiai, Geometric Function Theory in Several Complex Variables, Transl. Math. Mon.80, Amer. Math. Soc., Providence, R.I. 1990 · Zbl 0713.32001
[10] [Oc77] T. Ochiai: On holomorphic curves in algebraic varieties with ample irregularity, Invent. Math.443 (1977), 83-96 · Zbl 0374.32006
[11] [RW95] M. Ru, P.-M. Wong: Holomorphic curves in abelian and semi-abelian varieties, Preprint 1995
[12] [Si95] Y.-T. Siu: Hyperbolicity problems in function theory, in: ?Five Decades as a Mathematician and Educator on the 80th Birthday of Professor Yung-Chow Wong?, ed. K.-Y. Chan and M.-T. Liu, World Scientific 1995, pp.409-513
[13] [Va31] G. Valiron: Sur la dérivée des fonctions algébroides, Bull. Soc. Math. France59 (1931), 17-39
[14] [Vo92] P. Vojta: Integral points on sub-varieties of semi-abelian varieties, Preprint 1992
[15] [W80] P.-M. Wong: Holomorphic mappings into Abelian varieties, Amer. J. Math.102 (1980), 493-501 · Zbl 0439.32010
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