Geodesic
Compact complex manifolds with numerically effective cotangent bundles
Documenta mathematica, Tome 2 (1997), pp. 183-193
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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. 
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     author = {Henrik Kratz},
     title = {Compact complex manifolds with numerically effective cotangent bundles},
     journal = {Documenta mathematica},
     pages = {183--193},
     year = {1997},
     volume = {2},
     doi = {10.4171/dm/27},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/27/}
}
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Henrik Kratz. Compact complex manifolds with numerically effective cotangent bundles. Documenta mathematica, Tome 2 (1997), pp. 183-193. doi: 10.4171/dm/27

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