The dimension of harmonic currents on foliated complex surfaces

Abstract

Let $\mathcal{F}$ be a singular holomorphic foliation on an algebraic complex surface $S$, with hyperbolic singularities and no foliated cycle. We prove a formula for the transverse Hausdorff dimension of the unique harmonic current, involving the Furstenberg entropy and the Lyapunov exponent. In particular, we extend Brunella's inequality to every holomorphic foliation $\mathcal{F}$ on ${\mathbb{P}}^2$: if $\mathcal{F}$ has degree $d\ge 2$, then the Hausdorff dimension of its harmonic current is smaller than or equal to $\frac{d-1}{d+2}$, in particular the harmonic current is singular with respect to the Lebesgue measure. We also show that the Hausdorff dimension of the harmonic current of the Jouanolou foliation of degree $2$ is equal to $1/4$, and that the same property holds for topologically conjugate foliations on ${\mathbb{P}}^2$.