On closed currents invariant by holomorphic foliations, I

This paper introduces a dynamical approach to the study of singular holomorphic foliations carrying an invariant positive closed current on a compact complex surface. The approach consists of making a global sense of the dynamics of a special real $1$-dimensional oriented singular foliation well-known from the theory associated to the Godbillon–Vey invariant for codimension $1$ foliations. The problem then quickly splits in two genuinely different cases, to be separately treated, depending on whether or not the trajectories of the mentioned real foliation are all of “finite length”. The paper then continues by considering the case in which the support of the current in question contains at least one such trajectory having infinite length. By exploiting the contracting properties of the holonomy of $\mathcal{F}$ over the trajectories in question, we manage to prove in particular that $\mathcal{F}$ leaves an algebraic curve invariant.