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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.

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