Singular holomorphic foliations by curves II: Negative Lyapunov exponent

Abstract

Let \Fc be a holomorphic foliation by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: (1) the singular points of \Fc are all hyperbolic; (2) \Fc is Brody hyperbolic. Then we establish cohomological formulas for the Lyapunov exponent and the Poincar\'e mass of an extremal positive \ddc-closed current tangent to \Fc. If, moreover, there is no nonzero positive closed current tangent to \Fc, then we show that the Lyapunov exponent \chi(\Fc) of \Fc, which is, by definition, the Lyapunov exponent of the unique normalized positive \ddc-closed current tangent to \Fc, is a strictly negative real number. As an application, we compute the Lyapunov exponent of a generic foliation with a given degree in $\mathbb P^2.$