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Simultaneous uniformization for the leaves of projective foliations by curves. (English) Zbl 0821.32027

Let \({\mathcal F}\) be a foliation of degree \(\geq 2\) defined by curves on \(\mathbb{P}^ n_ \mathbb{C}\) whose singularities \(S({\mathcal F})\) have Milnor number 1. In this situation a \(C^ 2\) hermitian metric on \(\mathbb{P}^ n_ \mathbb{C} \backslash S ({\mathcal F})\) can be constructed which induces negative Gaussian curvature on the leaves of \({\mathcal F}\). It is proved that in this case all leaves of the foliation are uniformized by the unit disc. The set of uniformizations of the leaves is paracompact. Consequences concerning the nonexistence of vanishing cycles etc. are given.

MSC:

32S65 Singularities of holomorphic vector fields and foliations
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