Consider a foliation in the projective plane admitting a unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. With T.C Dinh, we showed that there is a unique positive ddc-closed (1,1)-current of mass 1 which is directed by the foliation. This is the current of integration on the invariant curve. A unique ergodicity theorem for the distribution of leaves follows: for any leaf L , appropriate averages on L converge to the current of integration on the invariant curve (although generically the leaves are dense). The result uses our theory of densities for currents. It extends to Foliations on Kähler surfaces.

I will describe recent extensions, with T.C Dinh and V.A Nguyen, dealing with foliations on compact Kähler surfaces. If the foliation, has only hyperbolic singularities and does not admit a transverse measure, in particular no invariant compact curve, then there exists a unique positive ddc-closed (1,1)-current of mass 1 which is directed by the foliation( it's like uniqueness of invariant measure for discrete dynamical systems). This improves on previous results, with J.E Fornaess, for foliations (without invariant algebraic curves) on the projective plane. The new proof uses an extension of the theory of densities to tensor products of positive ddc-closed currents.