A characterization of uniruled compact K\"ahler manifolds

TL;DR

This paper extends algebraicity and integrability criteria to foliations on compact K"ahler manifolds, establishing that such a manifold is uniruled if and only if its canonical bundle is not pseudoeffective.

Contribution

It adapts Bost's algebraicity characterization and extends Campana-Pe2un and Druel's criteria to foliations on compact K"ahler manifolds, providing a new characterization of uniruledness.

Findings

A compact K"ahler manifold is uniruled iff its canonical bundle is not pseudoeffective.

Extension of algebraic integrability criteria to foliations on K"ahler manifolds.

Application of adapted algebraicity characterization to establish uniruledness criterion.

Abstract

We adapt Bost's algebraicity characterization to the situation of a germ in a compact K\"ahler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-P\u{a}un and of Druel to foliations on compact K\"ahler manifolds. As an application, we prove that a compact K\"ahler manifold is uniruled if and only if its canonical line bundle is not pseudoeffective.