Singular holomorphic foliations by curves I: Integrability of holonomy   cocycle in dimension 2
  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. 
Viet-Anh Nguyen
math.DS math.CV math.DG