Integrability of holonomy cocycle for singular holomorphic foliations

We study the holonomy cocycle associated with a holomorphic Riemann surface foliation with a finite number of linearizable singularities. By accelerating the leafwise Poincare metric near the singularities, we show that the holonomy cocycle is integrable with respect to each harmonic probability measure directed by the foliation. As an application we establish the existence of Lyapunov exponents as well as an Oseledec decomposition for this cocycle.