Singular holomorphic foliations by curves I: Integrability of holonomy cocycle in dimension 2

Abstract: We study the holonomy cocycle H of a holomorphic foliation \Fc by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: 1) its singularities E are all hyperbolic; 2) there is no holomorphic non-constant map \C\to X such that out of E the image of \C is locally contained in a leaf. Let T be a harmonic current tangent to \Fc which does not give mass to any invariant analytic curve. Using the leafwise Poincar\'e metric, we show that H is integrable with respect to T. Consequently, we infer the existence of the Lyapunov exponent function of T.