Logarithmic Jet Bundles and Applications

Hyperbolic complex manifolds have been studied extensively during the last 30 years (see, for example, [10], [11]). However, it is still an important problem in hyperbolic geometry to understand the algebro-geometric and the differential-geometric meaning of hyperbolicity. The use of jet bundles has become a powerful tool to attack this problem. For example, Green and Griffiths ([5]) explained 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. Siu and Yeung ([22]) clarified this approach. Moreover, they gave a Second Main Theorem for divisors in abelian varieties, which was, very recently, clarified and generalized to the case of semi-tori by Noguchi, Winkelmann and Yamanoi ([18]). Demailly ([2]) presented a new construction of projective jets and pseudo-metrics on them which realizes directly the approach to Bloch’s theorem given in [5]. These projective jets are closer to the geometry of holomorphic curves than the usual jets, since the action of the group of reparametrizations of germs of curves, which is geometrically redundant, is divided out. Using these pseudometrics on projective jets, Demailly and El Goul ([3], see also Mc Quillan ([13])) were able to show that a (very) generic surface X in P of degree d ≥ 21 is Kobayashi hyperbolic. As a corollary one obtains that the complement of a (very) generic curve in P of degree d ≥ 21 is hyperbolic and hyperbolically embedded, a result first proved by Siu and Yeung ([20]) for much higher degree, using jet bundles and value distribution theory. In both papers this quasi-projective case is treated by proving hyperbolicity of a branched cover over the compactification. However, it is desirable to have also a direct approach to deal with quasiprojective varieties, since one can hope to get easier proofs and even better results . So one should also consider the case of logarithmic jet bundles. Noguchi ([16]) did this already for the case of the jet bundles used by Green-Griffiths. Via these bundles he generalized Bloch’s theorem to semi-abelian varieties. The main purpose of the present paper is to generalize Demailly’s construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. In sections 1 to 3, we establish this logarithmic generalization of Demailly’s construction explicitly via coordinates, just as Noguchi’s generalization of the jets used by Green-Griffiths. These explicit coordinates should be very useful for further applications. We also have another, more intrinsic way to obtain the same generalization in [4], which is much shorter, but does not give coordinates right away. In section 4 we prove the Ahlfors