Compact complex manifolds with numerically effective cotangent bundles

Abstract

We prove that a projective manifold of dimension $n=2$ or 3 and Kodaira dimension 1 has a numerically effective cotangent bundle if and only if the Iitaka fibration is almost smooth, i.e. the only singular fibres are multiples of smooth elliptic curves ($n=2$) resp. multiples of smooth Abelian or hyperelliptic surfaces ($n=3$). In the case of a threefold which is fibred over a rational curve the proof needs an extra assumption concerning the multiplicities of the singular fibres. Furthermore, we prove the following theorem: let $X$ be a complex manifold which is hyberbolic with respect to the Carathéodory-Reiffen-pseudometric, then any compact quotient of $X$ has a numerically effective cotangent bundle.