The Kobayashi pseudometric on algebraic manifold and a canonical fibration

Abstract

Given a compact complex manifold X of dimension n, we define a bimeromorphic invariant $\kappa_+(X)$ as the maximum p for which there is a saturated line subsheaf L of the sheaf of holomorphic p forms whose Kodaira dimension $\kappa (L)$ equals p. We call X special if $\kappa_+(X)=0$. We give some evidence that this condition characterizes X to have identically vanishing Kobayashi pseudometric. We use the well-known construction of F. Campana to give, for each projective X a canonical fibration $f: X\to Y$ holomorphic outside a proper subvariety of Y and whose general fibers are special. We show that the inherited orbifold structure on Y defined via the minimum multiplicity of those of the components of each fiber does not admit positive dimensional special sub-orbifolds through the general points of Y. We note that the Iitaka fibration or any rationally connected fibration of X is a natural factor of f and we show that this solves a general conjecture in Mori's classification program of algebraic varieties, namely, that an algebraic variety is either of general type, or (birationally) has a canonical fibration with positive dimensional special type fibers that factors through the Iitaka and rationally connected fibrations of X.