Numerical trivial foliations, Iitaka Fibrations and the numerical dimension

TL;DR

This paper explores the relationship between numerically trivial foliations, Iitaka fibrations, and the numerical dimension of pseudo-effective line bundles, providing criteria for abundance and new insights into their geometric structure.

Contribution

It generalizes the notion of numerically trivial foliations and links them to Iitaka fibrations, offering new criteria for the abundance of line bundles.

Findings

Leaves of the foliation have codimension at least the numerical dimension.

If Kodaira dimension equals numerical dimension, the Iitaka fibration is the numerically trivial foliation.

Provides a criterion for when a line bundle is not abundant.

Abstract

Modifying the notion of numerically trivial foliation of a pseudo-effective line bundle L introduced by the author in math.AG/0304312 it can be shown that the leaves of this foliation have codimension bigger or equal to the numerical dimension of L as defined by Boucksom, Demailly, Paun and Peternell, math.AG/0405285. Furthermore, if the Kodaira dimension of L equals its numerical dimension the Kodaira-Iitaka fibration is its numerically trivial foliation. Both statements together yield a sufficient criterion for L not being abundant.