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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.

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