Intrinsic pseudo-volume forms for logarithmic pairs

Abstract

We study an adaptation to the logarithmic case of the Kobayashi- Eisenman pseudo-volume form, or rather an adaptation of its variant defined by Claire Voisin, for which she replaces holomorphic maps by holomorphic K-correspondences. We define an intrinsic logarithmic pseudo-volume form X;D for every pair (X;D) consisting of a complex manifold X and a normal crossing Weil divisor D on X, the positive part of which is reduced. We then prove that X;D is generically non- degenerate when X is projective and KX +D is ample. This result is analogous to the classical Kobayashi-Ochiai theorem. We also show the vanishing of X;D for a large class of log-K-trivial pairs, which is an important step in the direction of the Kobayashi conjecture about infinitesimal measure hyperbolicity in the logarithmic case. Resume (Pseudo-formes volumes intrinseques pour les paires logarithmiques)