On foliations with nef anti-canonical bundle

TL;DR

This paper investigates the properties of the anti-canonical bundle in holomorphic foliations on complex projective manifolds, establishing conditions under which it cannot be nef and big, and exploring cases with maximal Kodaira dimension.

Contribution

It proves that the anti-canonical bundle cannot be nef and big for regular foliations or those with a compact leaf, and studies nef anti-canonical bundles with maximal Kodaira dimension.

Findings

Anti-canonical bundle not nef and big for regular foliations

Anti-canonical bundle not nef and big for foliations with a compact leaf

Analysis of nef anti-canonical bundles with maximal Kodaira dimension

Abstract

In this paper we prove that the anti-canonical bundle of a holomorphic foliation $\mathcal{F}$ on a complex projective manifold cannot be nef and big if either $\mathcal{F}$ is regular, or $\mathcal{F}$ has a compact leaf. Then we address codimension one regular foliations whose anti-canonical bundle is nef with maximal Kodaira dimension.