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Foliations in \(\mathbb{C} P(n)\): About hyperbolic holonomy for minimal exceptional sets. (Feuilletages de \(\mathbf C\mathbf P(n)\): De l’holonomie hyperbolique pour les minimaux exceptionnels.) (French) Zbl 0782.32023
Does there exist a holomorphic foliation \({\mathcal F}\) of codimension \(l\) in \(\mathbb{C} P(n)\) with a minimal exceptional set, i.e. with a leaf \(L\) whose closure \(\overline L\) does not contain any singular point of \({\mathcal F}\)? The answer is not known. However, the authors show: given a holomorphic foliation \({\mathcal F}\) of codimension \(l\) in \(\mathbb{C} P(n)\) with a leaf \(L\) such that \(\overline L\) is disjoint from the singular set of \({\mathcal F}\), there exists a loop in a leaf contained in \(\overline L\) with contracting hyperbolic holonomy.

32S65 Singularities of holomorphic vector fields and foliations
Full Text: DOI Numdam EuDML
[1] C. Camacho, A. Lins, P. Sad, Minimal sets of foliations on complex projective spaces,Publ. Math. I.H.E.S.,68 (1988), 187–203. · Zbl 0682.57012
[2] R. Langevin, H. Rosenberg, Quand deux sous-variétés sont forcées par leur géométrie à se rencontrer (à paraître auBulletin de la S.M.F.).
[3] J. F. Mattéi, R. Moussu, Holonomie et intégrales premières,Ann. Sci. Ec. Norm. Sup., Série 4, 13 (1980), 469–523.
[4] R. Narasimhan, Introduction to the theory of analytic spaces,Springer Lect. Notes in Math., no 25 (1966). · Zbl 0168.06003
[5] R. Poincaré, Mémoires sur les courbes définies par une équation différentielle,J. Math. Pures et Appl. (Série 3), 7 (1881), 375–422. · JFM 13.0591.01
[6] B. Raymond,Ensembles de Cantor et feuilletages, thèse de doctorat d’Etat, Paris XI (Orsay), 2 juin 1976.
[7] R. Sacksteder, Foliations and pseudo-groups,American Journal of Math.,87 (1965), 79–102. · Zbl 0136.20903 · doi:10.2307/2373226
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