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 Poincaré metric, we show that H is integrable with respect to T. Consequently, we infer the existence of the Lyapunov exponent function of T.

88 pages. In this fourth version we have added some minor corrections, Invent. math. (2017)

References in corpus (2)

• Unique ergodicity for foliations in P^2 with an invariant curve
• Geometric characterization of Lyapunov exponents for Riemann surface laminations

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