A refined Kodaira dimension and its canonical fibration

Abstract

Given a (meromorphic) fibration $f:X\to Y$ where $X$ and $Y$ are compact complex manifolds of dimensions $n$ and $m$, we define $L_f$ to be the invertible subsheaf of the sheaf of holomorphic $m$-forms of $X$ given by the saturation of $f^*K_Y$, where $K_Y$ is the canonical sheaf of $Y$. We define the Kodaira dimension of the orbifold base $(Y,f)$ by that of $L_f$ and call the fibration to be of general type if this dimension is the maximal $\dime Y$. We call $X$ special if it does not have a general type fibration with positive dimensional base and call $f$ special if its general fibers are. We note that Iitaka and rationally connected fibrations are special. Our main theorem is that any compact complex $X$ has a special fibration of general type which dominates any fibration of general type and factors through any special fibration of general type from $X$ in the birational category, thus unique in this category. Also, the fibration has positive dimensional fibers if $X$ is not of general type thus resolving a problem in Mori's program for varieties with negative Kodaira dimension. Our main theorem is stated in the full orbifold context of log pairs as in Mori's program via which we give an application to the Albanese map of projective manifolds with nef anticanonical bundle a part of which was derived using positive charateristic techniques by Qi Zhang.