Logarithmic Surfaces and Hyperbolicity
[Surfaces logarithmiques et hyperbolicité]
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.

DOI : 10.5802/aif.2307
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
Dethloff, Gerd 1 ; Lu, Steven S.-Y. 2

1 Université de Bretagne Occidentale UFR Sciences et Techniques Département de Mathématiques 6, avenue Le Gorgeu, BP 452 29275 Brest Cedex (France)
2 Université du Québec à Montréal Département de Mathématiques 201 av. du Président Kennedy Montréal H2X 3Y7 (Canada)
@article{AIF_2007__57_5_1575_0,
     author = {Dethloff, Gerd and Lu, Steven S.-Y.},
     title = {Logarithmic {Surfaces} and {Hyperbolicity}},
     journal = {Annales de l'Institut Fourier},
     pages = {1575--1610},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {5},
     year = {2007},
     doi = {10.5802/aif.2307},
     zbl = {1142.14024},
     mrnumber = {2364143},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2307/}
}
TY  - JOUR
AU  - Dethloff, Gerd
AU  - Lu, Steven S.-Y.
TI  - Logarithmic Surfaces and Hyperbolicity
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 1575
EP  - 1610
VL  - 57
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2307/
DO  - 10.5802/aif.2307
LA  - en
ID  - AIF_2007__57_5_1575_0
ER  - 
%0 Journal Article
%A Dethloff, Gerd
%A Lu, Steven S.-Y.
%T Logarithmic Surfaces and Hyperbolicity
%J Annales de l'Institut Fourier
%D 2007
%P 1575-1610
%V 57
%N 5
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2307/
%R 10.5802/aif.2307
%G en
%F AIF_2007__57_5_1575_0
Dethloff, Gerd; Lu, Steven S.-Y. Logarithmic Surfaces and Hyperbolicity. Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1575-1610. doi : 10.5802/aif.2307. http://archive.numdam.org/articles/10.5802/aif.2307/

[1] Barth, W.; Peters, C.; Ven, A. Van de Compact Complex Surfaces, Springer Verlag, 1984 | MR | Zbl

[2] Berteloot, F.; Duval, J. Sur l’hyperbolicité de certains complémentaires, Enseign. Math. II. Ser., Volume 47 (2001), pp. 253-267 | Zbl

[3] Brunella, M. Courbes entières et feuilletages holomorphes, Enseign. Math. II. Ser., Volume 45 (1999), pp. 195-216 | MR | Zbl

[4] Brunella, M. Birational geometry of foliations, First Latin American Congress of Mathematicians, IMPA (2000) | MR | Zbl

[5] Buzzard, G.; Lu, S. Algebraic surfaces holomorphically dominable by 2 , Inventiones Math., Volume 139 (2000), pp. 617-659 | DOI | MR | Zbl

[6] Buzzard, G.; Lu, S. Double sections, dominating maps and the Jacobian fibration, Amer. J. Math., Volume 122 (2000), pp. 1061-1084 | DOI | MR | Zbl

[7] Catanese, F. On the moduli spaces of surfaces of general type, J. Diff. Geom., Volume 19 (1984), pp. 483-515 | MR | Zbl

[8] Deligne, P. Théorie de Hodge II, Publ. Math. IHÉS, Volume 40 (1971), pp. 5-57 | Numdam | MR | Zbl

[9] Demailly, J.-P.; Goul, J. El Hyperbolicity of Generic Surfaces of High Degree in Projective 3-Spaces, Amer. J. Math., Volume 122 (2000), pp. 515-546 | DOI | MR | Zbl

[10] Dethloff, G.; Lu, S. Logarithmic jet bundles and applications, Osaka J. Math., Volume 38 (2001), pp. 185-237 | MR | Zbl

[11] Dethloff, G.; Schumacher, G.; Wong, P. M. Hyperbolicity of the complements of plane algebraic curves, Amer. J. Math., Volume 117 (1995), pp. 573-599 | DOI | MR | Zbl

[12] Dethloff, G.; Schumacher, G.; Wong, P. M. On the hyperbolicity of complements of curves in algebraic surfaces: The three component case, Duke Math. J., Volume 78 (1995), pp. 193-212 | DOI | MR | Zbl

[13] Goul, J. El Logarithmic jets and hyperbolicity, Osaka J. Math., Volume 40 (2003), pp. 469-491 | MR | Zbl

[14] Green, M. The hyperbolicity of the complement of 2n+1 hyperplanes in general position in n and related results, Proc. Amer. Math. Soc., Volume 66 (1977), pp. 109-113 | MR | Zbl

[15] Hartshorne, R. Algebraic Geometry, Graduate Texts in Math., Volume 52, Springer Verlag, New York, 1977 | MR | Zbl

[16] Iitaka, S. Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA, Volume 23 (1976), pp. 525-544 | MR | Zbl

[17] Iitaka, S. Algebraic Geometry, Graduate Texts in Math., Volume 76, Springer Verlag, New York, 1982 | MR | Zbl

[18] Kawamata, Y. Addition formula of logarithmic Kodaira dimension for morphisms of relative dimension one, Proc. Alg. Geo. Kyoto, Kinokuniya, Tokyo (1977), pp. 207-217 | MR | Zbl

[19] Kawamata, Y. Characterization of Abelian Varieties, Comp. Math., Volume 43 (1981), pp. 253-276 | Numdam | MR | Zbl

[20] Kawamata, Y.; Viehweg, E. On a characterization of an abelian variety in the classification theory of algebraic varieties, Comp. Math., Volume 41 (1980), pp. 355-360 | Numdam | MR | Zbl

[21] McQuillan, M. Non commutative Mori theory (Preprint)

[22] McQuillan, M. Diophantine approximation and foliations, Publ. Math. IHÉS, Volume 87 (1998), pp. 121-174 | Numdam | MR | Zbl

[23] Nishino, T.; Suzuki, M. Sur les Singularités Essentielles et Isolées des Applications Holomorphes à Valeurs dans une Surface Complexe, Publ. RIMS, Volume 16 (1980), pp. 461-497 | DOI | MR | Zbl

[24] Noguchi, J. Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties, Nagoya Math. J., Volume 83 (1981), pp. 213-223 | MR | Zbl

[25] Noguchi, J.; Ochiai, T. Geometric function theory in several complex variables, Transl. Math. Monographs, Amer. Math. Soc., 1990 | MR | Zbl

[26] Noguchi, J.; Winkelmann, J. Holomorphic curves and integral points off divisors, Math. Z., Volume 239 (2002), pp. 593-610 | DOI | MR | Zbl

[27] Noguchi, J.; Winkelmann, J.; Yamanoi, K. The second main theorem for holomorphic curves into semi-abelian varieties, Acta Math., Volume 188 (2002), pp. 129-161 | DOI | MR | Zbl

[28] Noguchi, J.; Winkelmann, J.; Yamanoi, K. Degeneracy of holomorphic curves into algebraic varieties (2005) (Preprint)

[29] Rousseau, E. Hyperbolicité du complémentaire d’une courbe: le cas de deux composantes, CRAS, Sér. I (2003), pp. 635-640 | Zbl

Cité par Sources :