Foliations whose first Chern class is nef

TL;DR

This paper studies foliations on projective manifolds with nef anti-canonical class, showing they induce a locally trivial fibration with a foliation of trivial canonical class on the base, under certain regularity or compact leaf conditions.

Contribution

It establishes a structure theorem for such foliations, demonstrating they are pullbacks of foliations with numerically trivial canonical class via a locally trivial fibration.

Findings

Existence of a locally trivial fibration for foliations with nef -K_{}

Foliations are pullbacks of trivial canonical class foliations on the base

Structural classification under regularity or compact leaf assumptions

Abstract

Let $\mathcal{F}$ be a foliation on a projective manifold $X$ with $-K_{\mathcal{F}}$ nef. Assume that either $\mathcal{F}$ is regular, or it has a compact leaf. We prove that there is a locally trivial fibration $f\colon X\to Y$, and a foliation $\mathcal{G}$ on $Y$ with $K_{\mathcal{G}} \equiv 0$, such that $\mathcal{F} = f^{-1}(\mathcal{G})$.