Singularities for analytic continuations of holonomy germs of Riccati foliations

Alvarez, Sébastien; Hussenot, Nicolas

Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 331-376.

Dans cet article, nous étudions le problème d’extension analytique de germes d’holonomie de feuilletages algébriques. Plus précisément, nous démontrons que pour un feuilletage de Riccati associé à une structure projective branchée sur une surface de type fini qui est non-élémentaire et parabolique, tous les germes d’holonomies entre une fibre et la section holomorphe du fibré vertical correspondante sont conduits vers une singularité par presque tout chemin géodésique développé. Nous étudions en détail la distribution de ces singularités et prouvons en particulier qu’elles forment une partie dense et indénombrable de l’ensemble limite. Cela redonne une réponse négative à une conjecture de Loray en utilisant une méthode complètement différente : l’étude ergodique du flot géodésique feuilleté initiée.

In this paper we study the problem of analytic extension of holonomy germs of algebraic foliations. More precisely we prove that for a Riccati foliation associated to a branched projective structure over a finite type surface which is non-elementary and parabolic, all the holonomy germs between a fiber and the corresponding holomorphic section of the bundle are led to singularities by almost every developed geodesic ray. We study in detail the distribution of these singularities and prove in particular that they form a dense uncountable subset of the limit set. This gives another negative answer to a conjecture of Loray using a completely different method, namely the ergodic study of the foliated geodesic flow.

Classification : 37D40, 37F75, 34M15, 32D15
Keywords: Riccati foliation, analytic continuation, foliated geodesic flow, Lyapunov exponents
Mot clés : feuilletages de Riccati, extensions analytiques, flot géodésique feuilleté, exponants de Lyapunov

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