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On foliations with nef anti-canonical bundle. (English) Zbl 1388.14056
Let $$X$$ be a complex projective manifold, and let $$\mathcal F \subset T_X$$ be a foliation, i.e. an integrable saturated subsheaf. In analogy to the classification of projective manifolds via the MMP, one expects that the anticanonical divisor $$-K_{\mathcal F} = \det(\mathcal F)$$ of the foliation encodes an important part of its geometry. For Fano foliations, i.e. the case when $$-K_{\mathcal F}$$ is ample, this has been studied in detail by C. Araujo and the author [Math. Ann. 360, No. 3–4, 769–798 (2014; Zbl 1396.14035); Adv. Math. 238, 70–118 (2013; Zbl 1282.14085)]. In this paper he considers the more general situation where $$-K_{\mathcal F}$$ is nef. Suppose morever that the foliation $$\mathcal F$$ is regular or has at least one compact leaf. The first result of this paper says that the divisor $$-K_{\mathcal F}$$ is not nef and big, thereby generalising a well-known theorem of Y. Miyaoka on the relative canonical bundle of a fibration over a curve [Comment. Math. Univ. St. Pauli 42, No. 1, 1–7 (1993; Zbl 0813.14001)]. For a codimension one foliation this immediately implies that the Kodaira dimension of $$-K_{\mathcal F}$$ is at most equal to its rank. For foliations of arbitrary codimension the same property is shown under the stronger assumption that the Kodaira dimension of $$-K_{\mathcal F}$$ is equal to its numerical dimension. Moreover one has $$\kappa(-K_{\mathcal F})=\text{rk}(\mathcal F)$$ if and only if $$\mathcal F$$ is algebraically integrable. The proof is based on the construction of an almost proper map $$\varphi : X \dashrightarrow Y$$ such that $$\mathcal F = \varphi^{-1} \mathcal G$$ for some foliation $$\mathcal G \subset T_Y$$ that is purely transcendental.

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 37F75 Dynamical aspects of holomorphic foliations and vector fields
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