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Holomorphic curves on hyperplane sections of 3-folds. (English) Zbl 0951.14014

Summary: We prove a conjectured height inequality of Lang and Vojta [see S. Lang, in: Number Theory. III: Diophantine Geometry, Encycl. Math. Sci. 60 (1991; Zbl 0744.14012)] for holomorphic curves lying on generic hyperplane sections of 3-folds. As a consequence we deduce a conjecture of S. Kobayashi [“Hyperbolic manifolds and holomorphic mappings” (1970; Zbl 0207.37902)] that a generic hypersurface in \(\mathbb{P}^3_\mathbb{C}\) of sufficiently high degree is hyperbolic.

MSC:

14H25 Arithmetic ground fields for curves
14J30 \(3\)-folds
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
32A60 Zero sets of holomorphic functions of several complex variables
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