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Finiteness theorems and hyperbolic manifolds. (English) Zbl 0742.14018

The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88, 163-178 (1990).
The author applies hyperbolic analysis to the study of sections of complex algebraic families \(f\colon X\to S\), in order to prove finiteness theorems.
The first section contains a short survey on hyperbolic manifolds with basic references to S. Kobayashi [Hyperbolic manifolds and holomorphic mappings. New York: Marcel Dekker (1970; Zbl 0207.37902)] and S. Lang [Fundamentals of diophantine geometry. New York etc.: Springer-Verlag (1983; Zbl 0528.14013)].
In section 2 the author presents a new proof of Mordell’s conjecture [Manin’s theorem, cf. Yu. I. Manin, Transl., Ser. 2, Am. Math. Soc. 50, 189–234 (1966); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 27, 1395–1440 (1963; Zbl 0166.16901)] for algebraic curves over function fields based on splitting of a fundamental group homomorphism and the finiteness of their conjugacy classes.
The methods are applied to abelian varieties over function fields in order to prove the following two theorems:
Theorem 1 [M. Raynaud in Algebraic geometry, Proc. Jap.-Fr. Conf., Tokyo and Kyoto 1982, Lect. Notes Math. 1016, 1–19 (1983; Zbl 0525.14014)]: Let \(X\subset A\) be a projective subvariety of an abelian variety, defined over a function field \(\mathbb{C}(B)\). If \(X\) does not contain a translation of an abelian subvariety, then the set \(X(K)\) of rational points is “finite modulo trace”.
Theorem 2 [“Weak Lang conjecture”, cf. S. Lang, Introduction to complex hyperbolic spaces. New York etc.: Springer-Verlag (1987; Zbl 0628.32001)]: Let \(H\) be a hyperplane section in some projective embedding of \(A\). Assume that \(H\) does not contain a translation of an abelian subvariety. Then any set of “\(S\)-integer points” on \(A\) is “ finite modulo trace”.
In the appendix the author presents a list of remarkable analogies around loops on complex manifolds and closed points on algebraic varieties over a finite field \(F\).
We remark that arithmetic analogues to the above theorems have been proved in the mean time by G. Faltings [Ann. Math. (2) 133, No. 3, 549–576 (1991; Zbl 0734.14007)]; see also L. Szpiro [in Sémin. Bourbaki, Vol. 1989/90, 42ème année, Astérisque 189/190, Exposé 729, 429–446 (1990; Zbl 0746.14010)].
[For the entire collection see Zbl 0717.00010.]

MSC:

14G05 Rational points
14H10 Families, moduli of curves (algebraic)
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
14K10 Algebraic moduli of abelian varieties, classification
32J15 Compact complex surfaces
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