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Logarithmic Vector Fields and Hyperbolicity

Published online by Cambridge University Press:  11 January 2016

Erwan Rousseau
Erwan Rousseau*
Affiliation:
Département de Mathématiques IRMA, Université Louis Pasteur, 7, rue René Descartes, 67084 Strasbourg Cedex, France rousseau@math.u-strasbg.fr
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Abstract

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Using vector fields on logarithmic jet spaces we obtain some new positive results for the logarithmic Kobayashi conjecture about the hyperbolicity of complements of curves in the complex projective plane. We are interested here in the cases where logarithmic irregularity is strictly smaller than the dimension. In this setting, we study the case of a very generic curve with two components of degrees d 1 ≤ d 2 and prove the hyperbolicity of the complement if the degrees satisfy either d 1 ≥ 4, or d 1 = 3 and d 2 ≥ 5, or d 1 = 2 and d 2 ≥ 8, or d 1 = 1 and d 2 ≥ 11. We also prove that the complement of a very generic curve of degree d at least equal to 14 in the complex projective plane is hyperbolic, improving slightly, with a new proof, the former bound obtained by El Goul.

Keywords

32Q4514J70

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 195 , 2009 , pp. 21 - 40
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

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