Unique ergodicity for foliations on compact Kähler surfaces

Abstract

Let F be a holomorphic foliation by Riemann surfaces on a compact Kähler surface X. Assume that it is generic in the sense that all the singularities are hyperbolic, and the foliation admits no directed positive closed $(1,1)$-current. Then there exists a unique (up to a multiplicative constant) positive $d{d}^c$-closed $(1,1)$-current directed by F. This is a very strong ergodic property of F showing that all leaves of F have the same asymptotic behavior. Our proof uses an extension of the theory of densities to a class of non-$d{d}^c$-closed currents. This is independent of foliation theory and represents a new tool in pluripotential theory. A complete description of the cone of directed positive $d{d}^c$-closed $(1,1)$-currents is also given when F admits directed positive closed currents.