Unique Ergodicity for foliations on compact K\"ahler surfaces
Complex Variables
2019-04-23 v2 Dynamical Systems
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.
Keywords
Cite
@article{arxiv.1811.07450,
title = {Unique Ergodicity for foliations on compact K\"ahler surfaces},
author = {Tien-Cuong Dinh and Viet-Anh Nguyen and Nessim Sibony},
journal= {arXiv preprint arXiv:1811.07450},
year = {2019}
}
Comments
Main results improved, proofs simplified, presentation changed. 50 pages