# Singular holomorphic foliations by curves II: Negative Lyapunov exponent

## TL;DR

This paper establishes formulas for the Lyapunov exponent of certain holomorphic foliations on complex surfaces, showing it is negative under specific conditions, with applications to generic foliations in projective planes.

## Contribution

It provides cohomological formulas for Lyapunov exponents of hyperbolic holomorphic foliations and proves negativity of the exponent in specific cases, extending understanding of foliation dynamics.

## Key findings

- Lyapunov exponent is strictly negative for certain foliations
- Cohomological formulas relate Lyapunov exponent and Poincaré mass
- Application to generic foliations in projective planes

## 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 $\mathbb P^2.$