Uniformisation de l'espace des feuilles de certains feuilletages de codimension 1

TL;DR

This paper investigates the geometric structure of leaf spaces in certain singular codimension-one foliations on compact Kähler manifolds, demonstrating they can be equipped with a metric of non-positive curvature that may degenerate on invariant hypersurfaces.

Contribution

It introduces a method to endow leaf spaces with a non-positive curvature metric using currents with minimal singularities, under the pseudo-effectiveness assumption.

Findings

Leaf space admits a metric of constant non-positive curvature

Degeneration occurs on rigidly embedded invariant hypersurfaces

Method applies to singular foliations with pseudo-effective conormal bundle

Abstract

This paper deals with codimension one (may be singular) foliations on compact K\"alher manifolds whose conormal bundle is assumed to be pseudo-effective. Using currents with minimal singularities, we show that one can endow the space of leaves with a metric of constant non positive curvature wich may degenerate on a rigidly embedded invariant hypersurface.