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Hyperbolic fibre spaces and Mordell’s conjecture over function fields. (English) Zbl 0583.32061
S. Lang [Bull. Am. Math. Soc., New Ser. 80, 779-787 (1974; Zbl 0298.14014)] discussed the higher dimensional analogue of Mordell’s conjecture for curves of genus \(\geq 2\) in terms of hyperbolic manifolds and posed a relative formulation of the problem for algebraic families of hyperbolic varieties: If there are an infinite number of cross sections, then the family contains split subfamilies and almost all cross sections are due to constant ones.
In this paper the author gives some affirmative answer to the above problem (main theorem). Then, assuming the conditions of the main theorem, he gives an analogue of Mordell’s conjecture over function fields which was proved by J. I. Manin [Izv. Akad. Nauk SSSR, Ser. Mat. 27, 1395-1440 (1963; Zbl 0166.169)] and H. Grauert [Publ. Math., Inst. Hautes √Čtud. Sci. 25, 131-149 (1965; Zbl 0137.405)] who didn’t assume these conditions.
Reviewer: G.A.Soifer

MSC:
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32L05 Holomorphic bundles and generalizations
14J10 Families, moduli, classification: algebraic theory
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