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Foliations with vanishing Chern classes. (English) Zbl 1302.32033
Let \(X\) be a complex projective manifold such that the Chern classes \(c_1(T_X)\) and \(c_2(T_X)\) vanish. By a classical result of Bieberbach this implies that some finite étale cover of \(X\) is an abelian variety. In the paper under review the authors consider foliations \(\mathcal F \subset T_X\) such that the Chern classes \(c_1(\mathcal F)\) and \(c_2(\mathcal F)\) vanish under the additional hypothesis that \(X\) is not uniruled, i.e., is not covered by rational curves. In their earlier paper with F. Loray [“Singular foliations with trivial canonical class”, Preprint, arXiv:1107.1538], the authors showed that for a non-uniruled manifold the condition \(c_1(\mathcal F)=0\) already implies that the foliation \(\mathcal F\) is smooth and there exists a transverse foliation \(\mathcal F^\perp\) such that \(T_X = \mathcal F \oplus \mathcal F^\perp\). In this paper they prove that the additional vanishing of the second Chern class yields a decomposition of the universal cover \(\tilde X\) as a product \(\mathbb C^{\dim \mathcal F} \times Y\). For a foliation of codimension two, one can even find a finite étale cover of \(X\) by \(A \times Y\) where \(A\) is an abelian variety and \(Y\) has dimension at most two (cf. [F. Touzet, Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 657–670 (2008; Zbl 1166.32014)] for the codimension-one case). For foliations of codimension larger than two it is in general not possible to find such a finite étale cover by a product, but the authors prove that there exists a meromorphic map \(X \dashrightarrow Y\) such that the general fibre is an abelian variety \(A\) and \(\mathcal F|_A \subset T_A\) is a linear foliation. Finally if the foliation \(\mathcal F\) is maximal with respect to inclusion and both \(X\) and \(\mathcal F\) are defined over a finitely generated \(\mathbb Z\)-algebra \(R \subset \mathbb C\), one can characterise the existence of a decomposition after finite cover in terms of Frobenius stability for the reduction modulo \(p\) of the foliation.

MSC:
32S65 Singularities of holomorphic vector fields and foliations
37F75 Dynamical aspects of holomorphic foliations and vector fields
57R30 Foliations in differential topology; geometric theory
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