Projective manifolds whose tangent bundles are numerically effective
It was proved by \textit{S. Mori} [Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)] that \({\mathbb{P}}^ n\) is the only complex projective n- fold whose tangent bundle is ample. - The authors deal with the problem of classifying n-folds X whose tangent bundle is numerically effective (nef). In the case \(n=2\), they easily show that X is either \({\mathbb{P}}^ 2\), or \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\), or a semistable projective bundle over an elliptic curve or an étale cover of an abelian surface. - For \(n=3\), the authors prove that X is an étale cover of an almost homogeneous 3-fold and that any fiber of its Albanese reduction is rational homogeneous. The authors also discuss the general case and conjecture that if \(T_ X\) is nef then X has an étale cover whose Albanese map is a flat analytic bundle with rational homogeneous fibers. For the proof in case \(n=3\) they show that either X is an étale cover of an abelian 3-fold or has Kodaira dimension -\(\infty\). In the latter case, there exists a contraction \(X\to Y\) with dim(Y)\(\leq 2\). A detailed study of all possible cases for Y completes the classification.
scientific article; zbMATH DE number 4204552
scientific article