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Degeneracy of holomorphic curves into algebraic varieties. (English) Zbl 1135.32018

This paper under review deals with a degeneracy theorem for entire holomorphic curves into algebraic varieties. Let \(X\) be a complex. We denote by \(\overline q(X)\) the log irregularity of \(X\). Let \(\overline\kappa(X)\) be the log Kodaira dimension of \(X\).
The main theorem in the paper is as follows: Suppose that there exists a finite morphism \(\pi: X\to A\) onto a semi-Abelian variety \(A\). Let \(f: \mathbb{C}\to X\) be an entire holomorphic curve. If \(\overline\kappa(X)> 0\), then \(f\) is algebraically degenerate. The normalization of the Zariski closure of the image of \(f\) is a semi-Abelian variety that is a finite étale cover of the translation of a proper semi-Abelian subvariety of \(A\).
The following degeneracy theorem is obtained as a corollary: Let \(X\) be a complex algebraic variety whose quasi-Albanese map is a proper map. Suppose that \(\overline\kappa(X)> 0\) and \(\overline q(X)> \dim X\), then every entire holomorphic curve \(f: \mathbb{C}\to X\) is algebraically degenerate.
The authors also give some applications for the Kobayashi hyperbolicity problem. For related topics, see G. Dethloff and S. S.-Y. Steven [Ann. Inst. Fourier 57, No. 5, 1575–1610 (2007; Zbl 1142.14024)].

MSC:

32H30 Value distribution theory in higher dimensions
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds

Citations:

Zbl 1142.14024
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References:

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