Compact Complex Manifolds with Numerically Effective Cotangent Bundles

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 Carath\'{e}odory-Reiffen-pseudometric, then any compact quotient of $X$ has a numerically effective cotangent bundle.

1991 Mathematics Subject Classification: 32C10, 32H20

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