# Logarithmic Surfaces and Hyperbolicity

Gerd Dethloff^{[1]}; Steven S.-Y. Lu^{[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)

Annales de l’institut Fourier (2007)

- Volume: 57, Issue: 5, page 1575-1610
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topDethloff, Gerd, and Lu, Steven S.-Y.. "Logarithmic Surfaces and Hyperbolicity." Annales de l’institut Fourier 57.5 (2007): 1575-1610. <http://eudml.org/doc/10271>.

@article{Dethloff2007,

abstract = {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.},

affiliation = {Université de Bretagne Occidentale UFR Sciences et Techniques Département de Mathématiques 6, avenue Le Gorgeu, BP 452 29275 Brest Cedex (France); Université du Québec à Montréal Département de Mathématiques 201 av. du Président Kennedy Montréal H2X 3Y7 (Canada)},

author = {Dethloff, Gerd, Lu, Steven S.-Y.},

journal = {Annales de l’institut Fourier},

keywords = {Classification of logarithmic surfaces; quasi-Albanese; foliations; logarithmic surfaces; entire curves; Brody curves},

language = {eng},

number = {5},

pages = {1575-1610},

publisher = {Association des Annales de l’institut Fourier},

title = {Logarithmic Surfaces and Hyperbolicity},

url = {http://eudml.org/doc/10271},

volume = {57},

year = {2007},

}

TY - JOUR

AU - Dethloff, Gerd

AU - Lu, Steven S.-Y.

TI - Logarithmic Surfaces and Hyperbolicity

JO - Annales de l’institut Fourier

PY - 2007

PB - Association des Annales de l’institut Fourier

VL - 57

IS - 5

SP - 1575

EP - 1610

AB - 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.

LA - eng

KW - Classification of logarithmic surfaces; quasi-Albanese; foliations; logarithmic surfaces; entire curves; Brody curves

UR - http://eudml.org/doc/10271

ER -

## References

top- W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces, (1984), Springer Verlag Zbl0718.14023MR749574
- F. Berteloot, J. Duval, Sur l’hyperbolicité de certains complémentaires, Enseign. Math. II. Ser. 47 (2001), 253-267 Zbl1009.32015
- M. Brunella, Courbes entières et feuilletages holomorphes, Enseign. Math. II. Ser. 45 (1999), 195-216 Zbl1004.32011MR1703368
- M. Brunella, Birational geometry of foliations, First Latin American Congress of Mathematicians (2000), IMPA Zbl1073.14022MR1948251
- G. Buzzard, S. Lu, Algebraic surfaces holomorphically dominable by ${\u2102}^{2}$, Inventiones Math. 139 (2000), 617-659 Zbl0967.14025MR1738063
- G. Buzzard, S. Lu, Double sections, dominating maps and the Jacobian fibration, Amer. J. Math. 122 (2000), 1061-1084 Zbl0967.32015MR1781932
- F. Catanese, On the moduli spaces of surfaces of general type, J. Diff. Geom. 19 (1984), 483-515 Zbl0549.14012MR755236
- P. Deligne, Théorie de Hodge II, Publ. Math. IHÉS 40 (1971), 5-57 Zbl0219.14007MR498551
- J.-P. Demailly, J. El Goul, Hyperbolicity of Generic Surfaces of High Degree in Projective 3-Spaces, Amer. J. Math. 122 (2000), 515-546 Zbl0966.32014MR1759887
- G. Dethloff, S. Lu, Logarithmic jet bundles and applications, Osaka J. Math. 38 (2001), 185-237 Zbl0982.32022MR1824906
- G. Dethloff, G. Schumacher, P. M. Wong, Hyperbolicity of the complements of plane algebraic curves, Amer. J. Math. 117 (1995), 573-599 Zbl0842.32021MR1333937
- G. Dethloff, G. Schumacher, P. M. Wong, On the hyperbolicity of complements of curves in algebraic surfaces: The three component case, Duke Math. J. 78 (1995), 193-212 Zbl0847.32028MR1328756
- J. El Goul, Logarithmic jets and hyperbolicity, Osaka J. Math. 40 (2003), 469-491 Zbl1048.32016MR1988702
- M. Green, The hyperbolicity of the complement of $2n+1$ hyperplanes in general position in ${\mathbb{P}}_{n}$ and related results, Proc. Amer. Math. Soc. 66 (1977), 109-113 Zbl0366.32013MR457790
- R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52 (1977), Springer Verlag, New York Zbl0367.14001MR463157
- S. Iitaka, Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA 23 (1976), 525-544 Zbl0342.14017MR429884
- S. Iitaka, Algebraic Geometry, Graduate Texts in Math. 76 (1982), Springer Verlag, New York Zbl0491.14006MR637060
- Y. Kawamata, Addition formula of logarithmic Kodaira dimension for morphisms of relative dimension one, Proc. Alg. Geo. Kyoto (1977), 207-217, Kinokuniya, Tokyo Zbl0437.14018MR578860
- Y. Kawamata, Characterization of Abelian Varieties, Comp. Math. 43 (1981), 253-276 Zbl0471.14022MR622451
- Y. Kawamata, E. Viehweg, On a characterization of an abelian variety in the classification theory of algebraic varieties, Comp. Math. 41 (1980), 355-360 Zbl0417.14033MR589087
- M. McQuillan, Non commutative Mori theory
- M. McQuillan, Diophantine approximation and foliations, Publ. Math. IHÉS 87 (1998), 121-174 Zbl1006.32020MR1659270
- T. Nishino, M. Suzuki, Sur les Singularités Essentielles et Isolées des Applications Holomorphes à Valeurs dans une Surface Complexe, Publ. RIMS 16 (1980), 461-497 Zbl0506.32007MR594913
- J. Noguchi, Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties, Nagoya Math. J. 83 (1981), 213-223 Zbl0429.32003MR632655
- J. Noguchi, T. Ochiai, Geometric function theory in several complex variables, Transl. Math. Monographs (1990), Amer. Math. Soc. Zbl0713.32001MR1084378
- J. Noguchi, J. Winkelmann, Holomorphic curves and integral points off divisors, Math. Z. 239 (2002), 593-610 Zbl1011.32012MR1893854
- J. Noguchi, J. Winkelmann, K. Yamanoi, The second main theorem for holomorphic curves into semi-abelian varieties, Acta Math. 188 (2002), 129-161 Zbl1013.32010MR1947460
- J. Noguchi, J. Winkelmann, K. Yamanoi, Degeneracy of holomorphic curves into algebraic varieties, (2005) Zbl1135.32018
- E. Rousseau, Hyperbolicité du complémentaire d’une courbe: le cas de deux composantes, CRAS, Sér. I (2003), 635-640 Zbl1034.32017

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.