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The Mordell property of hyperbolic fiber spaces with noncompact fibers. (English) Zbl 0858.32021

Let \(({\mathcal X}, p,{\mathcal B})\) be a complex fibre space. This is said to have the Mordell property, if it has at most finitely many meromorphic sections except for the ones which come from trivial sections of meromorphically trivial fibre subspaces of \(({\mathcal X}, p,{\mathcal B})\) modulo base change.
In 1974, Lang conjectured (as an analog of Mordells conjecture), that algebraic families of compact hyperbolic complex spaces have the preceding property [cf. also Manin 1963, H. Grauert, Inst. Hautes Etud. Sci., Publ. Math. 25, 363-381 (1965; Zbl 0137.40503)]. J. Noguchi [Int. J. Math. 3, No. 2, 277-289 (1992; Zbl 0759.32016), correct. ibid., No. 5, 677 (1992; Zbl 0767.32014)] has shown this for the case of a fibre space with compact fibres, which is hyperbolically embedded into its compactification \((\overline{\mathcal X}, \overline p, \overline {\mathcal B})\) along \(\partial {\mathcal B}\). The paper under review gives a noncompact version of Noguchi’s theorem:
Let \(\overline {\mathcal X}\) and \(\overline {\mathcal B}\) be irreducible, reduced compact complex spaces and \(\overline p: \overline {\mathcal X} \to \overline {\mathcal B}\) a surjective holomorphic mapping. Let \({\mathcal X}\) and \({\mathcal B}\) be Zariski open subsets of \(\overline {\mathcal X}\) and \(\overline {\mathcal B}\), respectively and \(p = \overline p|{\mathcal X}\) the restriction over \({\mathcal X}\) which is a surjection from \({\mathcal X}\) to \({\mathcal B}\). We consider the fiber space \(({\mathcal X}, p,{\mathcal B})\) with hyperbolic fibres, which we call a hyperbolic fibre space. Let \(\partial_h {\mathcal X}\) be the union of all irreducible components of \(\partial {\mathcal X} = \overline {\mathcal X} - {\mathcal X}\) which are not contained in \(\overline p^{-1} (\partial {\mathcal B})\) where \(\partial {\mathcal B} = \overline {\mathcal B} - {\mathcal B}\).
Main Theorem: Let \(({\mathcal X}, p,{\mathcal B})\) and \((\overline {\mathcal X}, \overline p, \overline {\mathcal B})\) be as above. Assume that \(({\mathcal X}, p,{\mathcal B})\) is hyperbolically imbedded into \((\overline {\mathcal X}, \overline p, \overline {\mathcal B})\) along the fibers and that \(\overline {\mathcal X} - \partial_h {\mathcal X}\) is locally complete hyperbolic in \(\overline {\mathcal X}\). Then \(({\mathcal X}, p, {\mathcal B})\) is Mordellic.
Reviewer: M.Roczen (Berlin)

MSC:

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32L05 Holomorphic bundles and generalizations
14H15 Families, moduli of curves (analytic)
11G99 Arithmetic algebraic geometry (Diophantine geometry)
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