Logarithmic surfaces and hyperbolicity (Q2461181)

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 \textit{J. Noguchi, J. Winkelmann} and \textit{K. Yamanoi} [J. Math. Pures Appl. (9) 88, No. 3, 293--306 (2007; Zbl 1135.32018)].