Logarithmic Surfaces and Hyperbolicity

Dethloff, Gerd; Lu, Steven S.-Y.

Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1575-1610.

J. Noguchi a démontré en 1981 que toute courbe entière est algébriquement dégénérée dans une variété algébrique logarithmique ayant une irrégularité logarithmique strictement plus grande que sa dimension.

Nous nous intéressons ici à des variétés dont l’irrégularité logarithmique est égale à la dimension. Nous nous restreignons au cas des courbes de Brody, pour lequel nous obtenons une solution complète du problème en dimension 2 : toute courbe de Brody dans une surface logarithmique ayant une irrégularité logarithmique égale à $2$ et de dimension de Kodaira logarithmique égale à $2$ est algébriquement dégénérée.

Nous obtenons encore le même résultat pour les variétés de dimension de Kodaira logarithmique égale à 1, sous une condition très faible portant sur la factorisation de Stein de l’application quasi-Albanese de la surface logarithmique. Nous démontrons également, par un contre-exemple, que le résultat ne tient plus sans cette condition.

Nous prouvons finalement qu’une surface logarithmique ayant une irrégularité logarithmique égale à 2 admet un certain type de courbes entières algébriquement non dégénérées si et seulement si leur dimension de Kodaira logarithmique est égale à zéro ; nous donnons également une caractérisation de ce cas en termes de l’application quasi-Albanese.

In 1981 J. Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate.

In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: in a logarithmic surface with logarithmic irregularity $2$ and logarithmic Kodaira dimension $2$, any Brody curve is algebraically degenerate.

In the case of logarithmic Kodaira dimension 1, we still get the same result under a very mild condition on the Stein factorization map of the quasi-Albanese map of the log surface, but we show by giving a counter-example that the result is not true any more in general.

Finally we prove that a logarithmic surface having logarithmic irregularity 2 admits certain types of algebraically non degenerate entire curves if and only if its logarithmic Kodaira dimension is zero, and we also give a characterization of this case in terms of the quasi-Albanese map.

Classification : 14J29, 32Q45, 14K12, 14K20, 32Q57, 32H25, 53C12, 53C55
Keywords: Classification of logarithmic surfaces, quasi-Albanese, foliations
Mot clés : classification des surfaces logarithmiques, quasi-Albanese, feuilletages

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