Manifolds with nef contangent bundle (Q383711)

A conjecture of \textit{H.-H. Wu} and \textit{F. Zheng} [J. Differ. Geom. 61, No. 2, 263--287 (2002; Zbl 1071.53539)], after a theorem of \textit{K. Ueno} [Classification theory of algebraic varieties and compact complex spaces. Notes written in collaboration with P. Cherenack. Lecture Notes in Mathematics. 439. Berlin-Heidelberg-New York: Springer-Verlag. XIX (1975; Zbl 0299.14007)], predicts that a Kähler manifold \(X\) with nef cotangent bundle should, after an étale cover, have Iitaka fibration \(X\to Y\) with \(K_Y\) ample and the fibres being complex tori. This paper proves the conjecture under the extra assumptions that \(X\) is projective and \(K_X\) is semiample. The latter assumption is needed because still nobody has proved Reid's abundance conjecture, but the projectivity assumption is inherent in the proof.NEWLINENEWLINEUeno [loc. cit.] proved this result for \(X\) a submanifold of a complex torus, and also showed that if additionally \(X\) is projective then (after an étale cover) \(X\) is a product \(Y\times T\). The latter is false with the assumption only that \(\Omega_X\) nef, but it is shown here that if \(X\) is projective and \(\Omega_X\) is semiample (some symmetric power is globally generated) then after an étale cover we have \(X=Y\times A\), with \(K_Y\) ample and \(A\) an abelian variety.NEWLINENEWLINEThe proof uses structure theorems for fibrations \(\varphi: X\to Y\) for which both \(T_X\) and \(T^*_X\) are nef on any \(Z\) contracted by \(\varphi\) (one says that \(T_X\) is \(\varphi\)-numerically flat), proved using results of Demailly, Peternell and Schneider [\textit{J.-P. Demailly} et al., J. Algebr. Geom. 3, No. 2, 295--345 (1994; Zbl 0827.14027)] on manifolds with nef tangent bundle. These results are then applied to the Iitaka fibration, which has that property in this case. The extra condition that \(X\) is projective is needed to ensure that the map is equidimensional.