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A KAM phenomenon for singular holomorphic vector fields

Laurent Stolovitch (2005)

Publications Mathématiques de l'IHÉS

Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection...

A note on M. Soares’ bounds

Eduardo Esteves, Israel Vainsencher (2006)

Annales de l’institut Fourier

We give an intersection theoretic proof of M. Soares’ bounds for the Poincaré-Hopf index of an isolated singularity of a foliation of ℂℙ n .

Abelian integrals in holomorphic foliations.

Hossein Movasati (2004)

Revista Matemática Iberoamericana

The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of a fixed degree and with a center singularity in the projective space of dimension two. Also we calculate higher Melnikov functions under some...

Algebraic foliations defined by complete vector fields

Alvaro Bustinduy (2011)

Banach Center Publications

The aim of this note is to give a clearer and more direct proof of the main result of another paper of the author. Moreover, we give some complementary results related to R-complete algebraic foliations with R a rational function of type ℂ*.

An Algebraic Formula for the Index of a Vector Field on an Isolated Complete Intersection Singularity

H.-Ch. Graf von Bothmer, Wolfgang Ebeling, Xavier Gómez-Mont (2008)

Annales de l’institut Fourier

Let ( V , 0 ) be a germ of a complete intersection variety in n + k , n > 0 , having an isolated singularity at 0 and X be the germ of a holomorphic vector field having an isolated zero at 0 and tangent to V . We show that in this case the homological index and the GSV-index coincide. In the case when the zero of X is also isolated in the ambient space n + k we give a formula for the homological index in terms of local linear algebra.

Bounding the degree of solutions to Pfaff equations

Marco Brunella, Luis Gustavo Mendes (2000)

Publicacions Matemàtiques

We study hypersurfaces of complex projective manifolds which are invariant by a foliation, or more generally which are solutions to a Pfaff equation. We bound their degree using classical results on logarithmic forms.

Chern numbers of a Kupka component

Omegar Calvo-Andrade, Marcio G. Soares (1994)

Annales de l'institut Fourier

We will consider codimension one holomorphic foliations represented by sections ω H 0 ( n , Ω 1 ( k ) ) , and having a compact Kupka component K . We show that the Chern classes of the tangent bundle of K behave like Chern classes of a complete intersection 0 and, as a corollary we prove that K is a complete intersection in some cases.

Cycles évanescents d’une fonction de Liouville de type f 1 λ 1 . . . f p λ p

Emmanuel Paul (1995)

Annales de l'institut Fourier

On construit un transport transverse aux fibres d’une fonction multivaluée de type f 1 λ 1 ... f p λ p ( λ i complexes), à l’origine de 2 . Ce transport est unique à isotopie près. On en déduit l’existence de voisinages réguliers dans lesquels les fibres sont toutes C difféomorphes (voire dans un cas quasi-homogène, analytiquement difféomorphes). On obtient également une généralisation de la notion de monodromie. On calcule enfin l’homologie évanescente de la fibre-type, en précisant le gradué qui lui est associé.

Dicritical logarithmic foliations.

Felipe Cano, Nuria Corral (2006)

Publicacions Matemàtiques

We show the existence of weak logarithmic models for any (dicritical or not) holomorphic foliation F of (C2,0) without saddle-nodes in its desingularization. The models are written in terms of a representative set of separatrices, whose equisingularity types are controlled by the Milnor number of the foliation.

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