Simultaneous uniformization for the leaves of projective foliations by curves

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In this paper we prove that, given a holomorphic foliation by curves on ℂP n , of degree ≥2, whose singularities have nondegenerate linear part, then there exists a hermitian metricg on ℂP n -S (S=singular set) which is complete and induces strictly negative Gaussian curvature on the leaves of the foliation (Theorem B). This implies, in particular, that all leaves of the foliation are uniformized by the unit disc and that the set of uniformizations of the leaves is paracompact (Theorem A). We obtain also some consequences concerning the non existence of vanishing cycles in the sense of Novikov, the equivalence of the existence of a parabolic element in the group of deck transformations of the leaf and of a separatrix in the leaf, etc...