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Logarithmic surfaces and hyperbolicity. (English) Zbl 1142.14024

The Green-Griffiths conjecture claims that every entire holomorphic curve in a variety of general type is algebraically degenerate, i.e., its image is contained in a proper closed subvariety of the given variety. The natural generalization to non-compact varieties is to assume that every entire curve to a (not necessarily compact) variety of maximal logarithmic Kodaira dimension is algebraically degenerate. The authors verify this for a special case. They show: If \(S\) is a surface with logarithmic irregularity \(2\) (the logarithmic irregularity is the dimension of the space of logarithmic one-forms and equals the dimension of the quasi-Albanese variety) and logarithmic Kodaira dimension \(2\), then every Brody curve is algebraically degenerate. (A Brody curve is an entire curve whose derivative is bounded.)
For surfaces with logarithmic irregularity 2 and logarithmic Kodaira dimension 1 it is shown that under a certain condition on the quasi-Albanese map (called “condition \((*)\)”) a similar result holds. Condition \((*)\) holds for example if all the fibers are compact. The authors give an example of a surface with logarithmic irregularity 2 and logarithmic Kodaira dimension 1 where \((*)\) does not hold and where indeed there exists a Zariski dense entire curve.
Related results have been obtained by J. Noguchi, J. Winkelmann and K. Yamanoi [J. Math. Pures Appl. (9) 88, No. 3, 293–306 (2007; Zbl 1135.32018)].

MSC:

14J29 Surfaces of general type
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32H25 Picard-type theorems and generalizations for several complex variables

Citations:

Zbl 1135.32018
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References:

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