Foliations with vanishing Chern classes

TL;DR

This paper classifies foliations with vanishing first and second Chern classes on non-uniruled projective manifolds, showing their universal cover splits and leaves are related to Abelian varieties, with implications for their arithmetic properties.

Contribution

It provides a structural description of such foliations, including the splitting of the universal cover and the nature of leaf closures, which was not previously understood.

Findings

Universal cover splits as a product

Zariski closures of leaves are Abelian varieties

Variation in leaf closures relates to arithmetic properties

Abstract

In this paper we aim at the description of foliations having tangent sheaf $T\mathcal F$ with $c_1(T\mathcal F)=c_2(T\mathcal F)=0$ on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as a product, and that the Zariski closure of a general leaf of $\mathcal F$ is an Abelian variety. It turns out that the analytic type of the Zariski closures of leaves may vary from leaf to leaf. We discuss how this variation is related to arithmetic properties of the tangent sheaf of the foliation.