  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 $\mathbb P^2.$
