Abstract
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...
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References
[B-K] J. Bland and M. Kalka, Complete metrics conformal to the hyperbolic disc, Proc. A.M.S., 97 (1986), 128–132.
[C] A. Candel, Uniformization of Surface Laminations. Ann. Sc. Ec. Norm. Sup. Field. Preprint.
[C-G] A. Candel and X. Gómez-Mont, Uniformization of the leaves of a Rational Vector Field. Preprint.
[CLS] C. Camacho, A. Lins Neto and P. Sad, Minimal Sets of Projcctive Foliations, Publ. Math. de I'IHES, #68.
[F] W. Fulton, Algebraic Curves. An Introduction to Algebraic Geometry, N. Y., W. A. Benjamin 1969.
[K] S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Marcel Dekker Inc., N. Y. 1970.
[L] S. Lang, Introduction to Complex Hyperbolic Spaces, Springer Verlag 1987.
[M] B. Maskit, Kleinian Groups, Springer Verlag 1987.
[N] S. P. Novikov, Topology of foliations, Trans. Moscow Math. Soc. 1965, pg. 268–304.
[V] A. Verjovsky, An Uniformization theorem for Holomorphic Foliations, Contemp. Math. 58(III) (1987) 233–253.
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Neto, A.L. Simultaneous uniformization for the leaves of projective foliations by curves. Bol. Soc. Bras. Mat 25, 181–206 (1994). https://doi.org/10.1007/BF01321307
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DOI: https://doi.org/10.1007/BF01321307