Intrinsic pseudo-volume forms for logarithmic pairs

TL;DR

This paper introduces an intrinsic logarithmic pseudo-volume form for pairs (X,D) in complex geometry, proving its non-degeneracy under certain conditions and exploring its vanishing in log-K-trivial cases, advancing understanding of hyperbolicity.

Contribution

It defines a new intrinsic pseudo-volume form for logarithmic pairs and proves its non-degeneracy and vanishing properties, extending classical results to the logarithmic setting.

Findings

_{X,D} is generically non-degenerate when X is projective and K_X+D is ample.

_{X,D} vanishes for a large class of log-K-trivial pairs.

The results extend classical hyperbolicity theorems to the logarithmic case.

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 \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.