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Compact complex manifolds with numerically effective tangent bundles. (English) Zbl 0827.14027
The main result of this fundamental article is: Let $$X$$ be a compact Kähler manifold with nef tangent bundle $$T_X$$. Moreover, let $$\widetilde X$$ be a finite étale cover of $$X$$ of maximum irregularity $$q = q (\widetilde X) = h^1 (\widetilde X, {\mathcal O}_{\widetilde X})$$. Then: $$\pi_1 (\widetilde X) \cong \mathbb{Z}^{2q}$$.
The albanese map $$\alpha : \widetilde X \to A (\widetilde X)$$ is a smooth fibration over a $$q$$-dimensional torus with nef relative tangent bundle.
The fibres of $$\alpha$$ are Fano manifolds with nef tangent bundles.
Here a line bundle $$L$$ on a compact complex manifold $$X$$ with a fixed hermitian metric $$\omega$$ is nef if, for every $$\varepsilon > 0$$, there exists a smooth hermitian metric $$h_\varepsilon$$ on $$L$$ such that the curvature satisfies $$\Theta_{h_\varepsilon} \geq - \varepsilon \omega$$. A bundle $$E$$ on $$X$$ is nef if the line bundle $${\mathcal O}_E (1)$$ on $$\mathbb{P} (E)$$ is nef. – Many other interesting and important results are contained in the article. It is proved that:
Let $$E$$ be a vector bundle on a compact Kähler manifold $$X$$.
If $$E$$ and $$E^*$$ are nef, then $$E$$ admits a filtration whose graded pieces are hermitian flat.
If $$E$$ is nef, then $$E$$ is numerically semi-positive.
Moreover, algebraic proofs are given for the result:
Any Moisheson manifold with nef tangent bundle is projective.
A compact Kähler $$n$$-fold with $$T_X$$ nef and $$c_1 (X)^n > 0$$ is Fano.
Further the two following classification results are given:
The smooth non-algebraic compact complex surfaces with nef tangent bundles are:
non-algebraic tori; Kodaira surfaces; Hopf surfaces.
Let $$X$$ be a non-algebraic three-dimensional compact Kähler manifold. Then $$T_X$$ is nef if and only if $$X$$, up to a finite étale cover, is either a torus or of the form $$\mathbb{P} (E)$$, where $$E$$ and $$E^*$$ are nef rank-2 vector bundles over a two-dimensional torus.

##### MSC:
 14J30 $$3$$-folds 14C20 Divisors, linear systems, invertible sheaves 32J17 Compact complex $$3$$-folds 14F35 Homotopy theory and fundamental groups in algebraic geometry 53C55 Global differential geometry of Hermitian and Kählerian manifolds 14E20 Coverings in algebraic geometry