Feuilletages holomorphes admettant une mesure transverse invariante

TL;DR

This paper investigates holomorphic foliations on compact Kähler manifolds with transverse invariant currents, establishing an alternative between the existence of an invariant hypersurface or a transverse invariant metric with constant curvature.

Contribution

It proves a dichotomy for such foliations, showing either the presence of an invariant hypersurface or a transverse invariant hermitian metric with constant curvature.

Findings

Existence of an invariant hypersurface under certain conditions.

Existence of a transverse invariant hermitian metric with constant curvature.

Dichotomy result for holomorphic foliations with invariant currents.

Abstract

Let $\cal{F}$ be a regular codimension 1 holomorphic foliation on a compact K\" ahler manifold. One assumes in addition that $\cal{F}$ possesses a transverse invariant positive current. The aim of this paper is to establish the following alternative: - There exists an invariant hypersurface. - The foliation admits a transverse invariant hermitian metric with constant curvature