Unique ergodicity for foliations in $$\mathbb {P}^2$$ with an invariant curve

Consider a foliation in the projective plane admitting a projective line as the unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. We show that there is a unique positive \({dd^c}\)-closed (1, 1)-current of mass 1 which is directed by the foliation and this is the current of integration on the invariant line. A unique ergodicity theorem for the distribution of leaves follows: for any leaf L, appropriate averages of L converge to the current of integration on the invariant line. The result uses an extension of our theory of densities for currents. Foliations on compact Kähler surface with one or more invariant curves are also considered.