Intrinsic pseudo-volume forms for logarithmic pairs

Details Intrinsic pseudo-volume forms for logarithmic pairs Intrinsic pseudo-volume forms for logarithmic pairs

2008-04-30

arxiv.org/abs/0804.4811v2

Bull. Soc. Math. France 138 (4): 543--582, 2010

Mathematics

Algebraic Geometry

Some typos corrected, e.g. in the statements of thm 1 and cor 2.11

Scientific paper

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 \Phi_{X,D} for every pair (X,D) consisting of a complex manifold X and a normal crossing Weil divisor, the positive part of which is reduced. We then prove that \Phi_{X,D} is generically non-degenerate when X is projective and K_X+D is ample. This result is analogous to the classical Kobayashi-Ochiai theorem. We also show the vanishing of \Phi_{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.

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