McQuillan, M. Holomorphic curves on hyperplane sections of 3-folds. (English) Zbl 0951.14014 Geom. Funct. Anal. 9, No. 2, 370-392 (1999). 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. Cited in 1 ReviewCited in 21 Documents 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 Keywords:height inequality; holomorphic curves; hyperplane sections of 3-folds Citations:Zbl 0744.14012; Zbl 0207.37902 × Cite Format Result Cite Review PDF Full Text: DOI