Positivity properties of the tangent bundle \(TX\) of an \(n\)-dimensional projective manifold \(X\) impose strong geometric restrictions on \(X\). Formerly a conjecture of Hartshorne, the celebrated result [S. Mori, Ann. Math. (2) 110, 593–606 (1979; Zbl 0423.14006)] showed that if \(TX\) is ample, then \(X\simeq\mathbb P^n\). The numerical effective (nef) property is a weakening of ampleness. Abelian varieties and rational homogeneous spaces are easy examples of manifolds with nef tangent bundles. The papers [F. Campana and T. Peternell, Math. Ann. 289, No. 1, 169–187 (1991; Zbl 0729.14032)] and [J.-P. Demailly et al., J. Algebr. Geom. 3, No. 2, 295–345 (1994; Zbl 0827.14027)] classify all such manifolds, generalizing Mori’s result.
Peternell asked whether the structure of \(X\) is similarly restricted if \(TX\) admits positive subsheaves \(\mathcal F\). If \(\mathcal F\) is ample, then \(X\simeq\mathbb P^n\). This was proved by M. Andreatta and J. A. Wiśniewski [Invent. Math. 146, No. 1, 209–217 (2001; Zbl 1081.14060)] and J. Liu [Nagoya Math. J. 233, 155–169 (2019; Zbl 1411.14054)]. If \(\mathcal F\) is strictly nef and locally free, [J. Liu et al, “Projective manifolds whose tangent bundle contains a strictly nef subsheaf”, Preprint, arXiv:2004.08507] (2020)] prove that \(X\) admits a \(\mathbb P^d\) bundle structure over a Brody hyperbolic manifolds.
This paper focuses on the case where \(X\) admits an almost nef regular foliation \(\mathcal F\subseteq TX\) with \(c_1(\mathcal F)\neq 0\). The basic example of a foliation is the relative tangent bundle of a smooth morphism from \(X\). Given that \(X\) is expected to be close to a \(\mathbb P^d\)-bundle, considering foliations instead of arbitrary subsheaves is a natural strengthening of the hypothesis.
The author proves that under the assumptions above, \(X\) is a smooth fibration \(f:X\to Y\) over some projective manifold and \(f\) has rationally connected fibers. Furthermore there exists \(\mathcal G\subseteq TY\) a numerically flat foliation such that \(\mathcal F\) is an extension of the sheaf pullback \(f^*\mathcal G\) by the relative \(T_{X/Y}\). With the appropriate terminology, \(\mathcal F\) is the foliation pullback \(f^{-1}\mathcal G\). Stronger positivity positivity assumptions on \(\mathcal F\) induce further restrictions on the geometry of \(X\) and \(f\). If \(\mathcal F\) is nef, then the fibers of \(f\) are Fanos. If \(\mathcal F\) is Viehweg-big, then the fibers are projective spaces and \(\mathcal F=T_{X/Y}\). Finally if \(\mathcal F\) is Viehweg-big and nef, then \(X\simeq\mathbb P^n\).
The proof uses a generalization of the Fujita decomposition to the setting of reflexive sheaves. This is introduced and studied here by the author. A wide range of results from the literature are also employed.
The paper is also concerned with a question of Druel regarding regular foliations with nef anticanonical bundle. The author proves that if \(\mathcal F\) is regular or has a compact leaf, then \(\kappa(X,-K_{\mathcal F})\leq\mathrm{rk}\,\mathcal F\). The equality case imposes further restrictions on \(\mathcal F\). It is an algebraically integrable foliation with rationally connected leaves, the anticanonical bundle \(-K_{\mathcal F}\) of the foliation is semiample, and the anticanonical bundle \(-K_F\) of the general leaf is big and semiample.
The proof of this theorem is more involved than that of the main result for almost nef regular foliations. It makes extensive use of results of Druel.
The writing style is generally efficient and the main ideas are easy to understand. To keep the paper short, several folklore results are used without being explicitly mentioned.