Codimension one regular foliations on rationally connected threefolds

TL;DR

This paper proves Touzet's conjecture that regular codimension one foliations on certain rationally connected threefolds are algebraically integrable with rationally connected leaves, extending previous results to a broader class of threefolds.

Contribution

It establishes the validity of Touzet's conjecture for codimension one foliations on threefolds with nef anti-canonical bundle, broadening the understanding of foliation integrability.

Findings

Touzet's conjecture holds for these threefolds.

Regular foliations are algebraically integrable with rationally connected leaves.

Extension of Druel's results to a new class of threefolds.

Abstract

In his work on birational classification of foliations on projective surfaces, Brunella showed that every regular foliation on a rational surface is algebraically integrable with rational leaves. This led Touzet to conjecture that every regular foliation on a rationally connected manifold is algebraically integrable with rationally connected leaves. Druel proved this conjecture for the case of weak Fano manifolds. In this paper, we extend this result showing that Touzet's conjecture is true for codimension one foliations on threefolds with nef anti-canonical bundle.