Numerically Trivial Foliations, Iitaka Fibrations and the Numerical Dimension

Modifying the notion of numerically trivial foliation of a pseudoeffective line bundle L introduced by the author in [Eck04a] (see also math.AG/0304312) it can be shown that the leaves of this foliation have codimension ≥ 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. 0. Introduction In their seminal paper [BDPP04] Boucksom, Demailly, Paun and Peternell introduced a numerical dimension for pseudo-effective (1, 1)-classes on compact Kähler manifolds generalizing the numerical dimension of nef line bundles on projective manifolds. For this purpose they used Boucksom’s moving intersection numbers [Bou02] which can be defined as follows: Definition 0.1. Let X be a compact Kähler manifold with Kähler form ω. Let α1, . . . , αp ∈ H (X,R) be pseudo-effective classes and let Θ be a closed positive current of bidimension (p, p). Then the moving intersection number (α1·. . .·αp·Θ)≥0 of the αi and Θ is defined to be the limit when ǫ > 0 goes to 0 of