Logarithmic Jets and Hyperbolicity

Abstract

We prove that the complement of a very generic curve of degree $d$ at least equal to 15 in the projective plane is hyperbolic in the sens of Kobayashi (here, the terminology ``very generic'' refers to complements of countable unions of proper algebraic subsets of the parameter space). We first consider the Dethloff and Lu's generalisation to the logarithmic situation of Demailly's jet bundles. We study their base loci for surfaces of log-general type in the same way as it was done in the compact case by Demailly and El Goul. With some condition on log-Chern classes, any entire holomorphic map to the surface can be lifted as a leaf of some foliation on a ramified covering. Then we obtain a logarithmic analogue of McQuillan's result on holomorphic foliations which permits to conclude. Using the logarithmic formalism, we even obtain some simplifications of the original proof in McQuillan's work.