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

Abstract

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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.

How to cite

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Dethloff, 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

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