Logarithmic Jet Bundles

Recently J. P. Demailly gave a natural desingularization of the quotient of jet bundles by the full reparametrization group. Using this, he was able to complete an approach by Green and Griiths to construct negatively curved jet pseudometrics on subvarieties of Abelian varieties to prove Bloch's theorem metrically. In the present paper we extend this technique to the logarithmic case, yielding a simple and eeective tool for studying the hyperbolic geometry of quasiprojective varieties. As a rst application, we use the technique to give a metric proof for the logarithmic version of Lang's conjecture concerning the hyper-bolicity of complements of divisors in a semi-abelian variety as well as for the corresponding big Picard theorem. 0 Introduction Complex hyperbolic manifolds are considered to be the generalizations of hyperbolic Riemann surfaces to higher dimensions. During the last 30 years they have been studied extensively (see, for example 10], 11]). However, it is still an important problem in hyperbolic geometry to understand the algebro-geometric and the diierential-geometric meaning of hyperbolicity. One of the most important papers in this direction is the paper 5] of Green and Griiths. It explains, for example, an approach to establish Bloch's theorem on the algebraic degeneracy of holomorphic maps into abelian varieties by constructing negatively curved pseudometrics on jet bundles and by applying Ahlfors' lemma. This paper has made geometrically transparent the basic problems in establishing Bloch's theorem. Unfortunately , the pseudometric constructed there does not always have strictly negative curvature (see also 2], 18]), which is necessary for applying the Ahlfors-Schwarz lemma. We remark however that Siu and Yeung ((18]) have 1