Unique Ergodicity for foliations on compact K\"ahler surfaces

TL;DR

This paper proves the uniqueness of a positive directed dc-closed (1,1)-current for generic holomorphic foliations on compact Ka9hler surfaces, revealing a strong ergodic property under certain conditions.

Contribution

It establishes the existence and uniqueness of a positive dc-closed (1,1)-current for generic foliations, extending density theory to non-dc-closed currents and describing the cone of such currents.

Findings

Unique positive dc-closed current exists for generic foliations.

Extension of density theory to non-dc-closed currents.

Complete description of the cone of directed positive dc-closed currents.

Abstract

Let \Fc be a holomorphic foliation by Riemann surfaces on a compact K\"ahler surface X. Assume it is generic in the sense that all the singularities are hyperbolic and that the foliation admits no directed positive closed (1,1)-current. Then there exists a unique (up to a multiplicative constant) positive \ddc-closed (1,1)-current directed by \Fc. This is a very strong ergodic property of \Fc. Our proof uses an extension of the theory of densities to a class of non-\ddc-closed currents. A complete description of the cone of directed positive \ddc-closed (1,1)-currents is also given when \Fc admits directed positive closed currents.