Transversely affine and transversely projective holomorphic foliations Annales scientifiques de l'É.N.S. 4 e série, tome 30, n o 2 (1997), p. 169-204 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1997, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Intuitively, a Liouvillian function on CP (n) is one which is obtained from rational functions by a finite process of integrations, exponentiations and algebraic operations. This paper is devoted to the study of foliations determined by polynomial 1-forms which have a Liouvillian first integral. Our main result states that, under some mild restrictions on the singularities of the foliation, such a foliation must be either a linear foliation or an exponent two Bernoulli foliation after some rational pull-back. This proves that the highest level of transcendence for the ordinary differential equations which can be integrated by the use of elementary functions is reached at the Riccati equations. † The singular holonomy groups are sorts of differential Galois groups of the foliation (see Theorem 3, the Structure theorem, in §3).

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Abstract. We prove that a holomorphic foliation of codimension k which is transverse to the fibers of a fibration and has a compact leaf with finite holonomy group is a Seifert fibration, i.e., has all leaves compact with finite holonomy. This is the case for C 1 -small deformations of a foliation where the original foliation exhibits a compact leaf and the base B of the fibration satisfies H 1 (B, R) = 0 and H 1 (B, GL(k, R)) = 0.

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Germs of holomorphic vector fields at the origin 0 ∈ C 2 and polynomial vector fields on C 2 are studied. Our aim is to classify these vector fields whose orbits have bounded geometry in a certain sense. Namely, we consider the following situations: (i) the volume of orbits is an integrable function, (ii) the orbits have sub-exponential growth, (iii) the total curvature of orbits is finite. In each case we classify these vector fields under some generic hypothesis on singularities. Applications to questions, concerning polynomial vector fields having closed orbits and complete polynomial vector fields, are given. We also give some applications to the classical theory of compact foliations.

We prove that if a holomorphic one-form in a neighborhood of a closed euclidian ball B 2n ⊂ C n , in the n-dimensional complex a ne space, deÿnes a distribution transverse to the boundary sphere S 2n−1 = @B 2n , then n is even and admits a sole singularity q ∈ B 2n . Moreover, this singularity is simple. ?