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This paper is devoted to the resolution of singularities of holomorphic vector fields and one-dimensional holomorphic foliations in dimension three, and it has two main objectives. First, within the general framework of one-dimensional foliations, we build upon and essentially complete the work of Cano, Roche, and Spivakovsky (2014). As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan–Panazzolo (2013) but proved by means of rather different methods. The other objective of this paper is to consider a special class of singularities of foliations containing, in particular, all the singularities of complete holomorphic vector fields on complex manifolds of dimension three. We then prove that a much sharper resolution theorem holds for this class of holomorphic foliations. This second result was the initial motivation for this paper. It relies on combining earlier resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations. Bibliography: 34 titles.

Статья посвящена изучению разрешений особенностей голоморфных векторных полей и одномерных слоений в размерности 3 и преследует две основные цели. Во-первых, в рамках общей теории одномерных слоений мы опираемся на работу Ф. Кано, К. Роша и М. Спиваковского (2014) и по существу завершаем еe. Как следствие, мы получаем общую теорему о разрешении, сравнимую с теоремой о разрешении Мак-Квиллана-Панаццоло (2013), но доказываемую с помощью совершенно иных методов. Вторая цель статьи состоит в рассмотрении специального класса особенностей слоений, содержащего, в частности, все особенности полных голоморфных векторных полей на комплексных многообразиях размерности 3. Мы доказываем, что для этого класса голоморфных слоений имеет место гораздо более сильная теорема о разрешении. Этот второй результат был первоначальной мотивацией данной статьи, и он основан на комбинации ранее полученных нами результатов о разрешении особенностей (общих) слоений и некоторых классических методов асимптотических разложений для решений дифференциальных уравнений. Библиография: 34 названия.

Given a pair of commuting holomorphic vector fields defined on a neighborhood of (0,0) in C^2, we discuss the problem of globalizing them as an action of C^2 on a suitable complex surface along with some related questions. A review of Palais' theory about globalization of local transformation groups is also included in our discussion., The first author was partially supported by FCT (Portugal) through the sabbatical grant SFRH/BSAB/135549/2018 and through CMAT (UID/MAT/00013/2013). The second author was partially supported by CIMI - Labex Toulouse - through the grant "Dynamics of modular vector fields, Dwork family, and applications". The third author was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020. Finally all the three authors benefited from CNRS (France) support through the PICS project "Dynamics of Complex ODEs and Geometry"., info:eu-repo/semantics/publishedVersion

We prove a global topological rigidity theorem for locally \begin{document}$ C^2 $\end{document} -non-discrete subgroups of \begin{document}$ {\rm {Diff}}^{\omega} (S^1) $\end{document} .

We show that holomorphic vector fields on ( C 3 , 0 ) have separatrices provided that they are embedded in a rank 2 representation of a two-dimensional Lie algebra. As an application of this result we show, in particular, that the second jet of a holomorphic vector field defined on a compact complex manifold M of dimension 3 cannot vanish at an isolated singular point provided that M carries more than a single holomorphic vector field.

We discuss the local dynamics of a subgroup of \({{\mathrm{Diff}}\, ({{\mathbb {C}}}^2, 0)}\) possessing locally discrete orbits as well as the structure of the recurrent set for more general groups. It is proved, in particular, that a subgroup of \({{\mathrm{Diff}}\, ({{\mathbb {C}}}^2, 0)}\) possessing locally discrete orbits must be virtually solvable. These results are of considerable interest in problems concerning integrable systems.

Consider a pseudogroup on $( \mathbb{C} , 0)$ generated by two local diffeomorphisms having analytic conjugacy classes a priori fixed in $\mathrm{Diff} \hspace{0.167em} ( \mathbb{C} , 0)$. We show that a generic pseudogroup as above is such that every point has a (possibly trivial) cyclic stabilizer. It also follows that these generic groups possess infinitely many hyperbolic orbits. This result possesses several applications to the topology of leaves of foliations, and we shall explicitly describe the case of nilpotent foliations associated to Arnold’s singularities of type ${A}^{2n+ 1} $.

In this paper we obtain some non-linear analogues of Schur's theorem asserting that a finitely generated subgroup of a linear group all of whose elements have finite order is, in fact, finite. The main result concerns groups of symplectomorphisms of certain manifolds of dimension $4$ including the torus $T^4$.

In this work we study the singularities at infinity of algebraic vector fields in dimension 2.These singularities will be classified under a mild assumption. The general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in Part II. Our main results are local and therefore can be carried over more general surfaces. Whereas we deal with ℂ -complete vector fields, the results also apply to ℝ-complete ones thanks to a theorem of Forstneric [7].

We investigate some global generic properties of the dynamics associated to non-Abelian free actions in certain special cases. The main properties considered in this paper are related to the existence of dense orbits, to ergodicity and to topological rigidity. We first deal with them in the case of conservative homeomorphisms of a manifold and C 1-diffeomorphisms of a surface. Groups of analytic diffeomorphisms of a manifold which, in addition, contain a Morse-Smale element and possess a generating set close to the identity are considered as well. From our discussion we also derive the existence of a rigidity phenomenon for groups of skew-products which is opposed to the phenomenon present in Furstenberg’s celebrated example of a minimal diffeomorphism that is not ergodic (cf. [Ma]).

We consider subgroups of C∞-diffeomorphisms of the circle 1 which act transitively on 4-tuples of points. We show, in particular, that these subgroups are dense in the group of homeomorphisms of 1 . A stronger result concerning C°°-approximations is obtained as well. The techniques employed in this paper rely on Lie algebra ideas and they also provide partial generalizations to the differentiable case of some results previously established in the analytic category.

In this work we study the singularities at infinity of algebraic vector fields in dimension 2.These singularities will be classified under a mild assumption. The general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in Part II. Our main results are local and therefore can be carried over more general surfaces. Whereas we deal with ℂ -complete vector fields, the results also apply to ℝ-complete ones thanks to a theorem of Forstneric [7].

We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space ℂℙ n for every dimension n≥2 and every degree d≥2. Precisely, we construct a foliation ℱ which is induced by a homogeneous vector field of degree d, has a finite singular set and all the regular leaves are dense in the whole of ℂℙ n . Moreover, ℱ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if ℱ is conjugate to another holomorphic foliation by a homeomorphism sufficiently close to the identity, then these foliations are also conjugate by a projective transformation. Finally, all these properties are persistent for small perturbations of ℱ.¶This is done by considering pseudo-groups generated on the unit ball 𝔹 n ⊆ℂ n by small perturbations of elements in Diff(ℂ n ,0). Under open conditions on the generators, we prove the existence of many pseudo-flows in their closure (for the C 0-topology) acting transitively on the ball. Dynamical features as minimality, ergodicity, positive entropy and rigidity may easily be derived from this approach. Finally, some of these pseudo-groups are realized in the transverse dynamics of polynomial vector fields in ℂℙ n .

In this note we extend to arbitrary dimensions a couple of results due respectively to Mattei-Moussu and to Camara-Scardua in dimension 2. We also provide examples of singular foliations having a Siegel-type singularity and answering in the negative the central question left open in the previous work of Camara-Scardua., Comment: This paper together with the paper "Discrete orbits, recurrence and solvable subgroups of Diff(C^2,0)" correspond to an expanded and accurate version of our previous preprint "Discrete orbits and special subgroups of Diff(C^n,0)" arXiv:1301.3716

We introduce a method to estimate the size of the domain of definition of the solutions of a meromorphic vector field on a neighborhood of its pole divisor. The corresponding techniques are, in a certain sense, quantitative versions of some well-known phenomena related to the presence of metrics with positive curvature. Several applications of these ideas are provided including a type of "confinement theorem" for solutions of complete polynomial vector fields on $\mathbb{C}^n$ and obstructions for certain (germs of) vector fields to be realized by a global holomorphic vector field on a compact K\"ahler manifold. As a complement a new approach to certain classical equations is proposed and detailed in the case of Halphen equations., Comment: This is a revised version of arXiv:1111.2365

Let Diff ω 0 ( T 2 ) denote the connected component of the identity of the group of real analytic diffeomorphisms of the torus of dimension 2 . We consider nilpotent subgroups Diff ω 0 ( T 2 ) and we show that they are metabelian. This implies in particular that any homomorphism from a subgroup of finite index of SL (n, Z ) (n≥4) to Diff ω 0 ( T 2 ) has a finite image.

Nous considerons les champs de vecteurs holomorphes de dimension 2 ayant deux valeurs propres non nulles a l'origine. Il est demontre que, un facteur inversible pres, tous ces champs sont semi-complets. On classifie aussi les singularites du type col-noeud (une valeur propre nulle) qui sont semi-completes. Finalement nous donnons des exemples de champs de vecteurs semi-complets avec une courbe de singularites et tels que le feuilletage singulier induit n'est pas linearisable.

We consider the non solvable subgroups of the group of real analytic diffeomorphisms of the circle which admit a finite generating set whose elements belong to an appropriate and fixed neighborhood of the identity. If G is such a group, we prove that there are non trivial local analytic vector fields which are a sort of “limit” of some local diffeomorphisms in G. Finally we apply these vector fields to prove, in particular, that either the group G is ergodic or it has a finite orbit. These vector fields also enable us to show that the dynamics of G is topologically rigid.

We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.

We consider closed positive currents invariant by a singular holomorphic foliation on an algebraic surface. We show that under some conditions the foliation must leave invariant an algebraic curve.

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We consider fibrations of genus 2 over complex surfaces. The purpose of this paper is primarily to provide a geometric description of the possible structures of the fibration on a neighborhood of a singular fiber. In particular it is shown that the "geometric data" of the singular fiber determines the fibration on its neighborhood up to a transversely holomorphic $C^{\infty}$-diffeomorphism. The method employed is quite flexible and it applies to good extent to fibrations of arbitrary genus., Comment: This is the final version, June 2011

We study singularities of semi-complete holomorphic vector fields in dimension 3. Under the very generic assumption that the singularities of the associated foliation can be resolved with a single blow-up, we establish Ghys's conjecture asserting that the second jet of the initial vector field cannot vanish at an isolated singularity. In fact, it results from our proof that, under the preceding assumption, it suffices to assume that the singular set of the mentioned vector field has codimension at least 2. Related to this question and, in fact, playing a central role in our proof, we also present a detailed picture of the semi-complete singularities of vector fields whose associated foliation has a single eigenvalue different from zero.

We consider measurable conjugacies between two non-solvable subgroups G_1, G_2 of the group of analytic diffeomorphisms of the circle which admit a finite generating set close to the identity. These measurable conjugacies are characterized by our main result which asserts that, up to finite coverings, they coincide almost everywhere with an analytic diffeomorphism of the circle.

Nous nous interessons a l'etude des champs de vecteurs holomorphes complets, notamment en dimension deux. Nous introduisons la notion de germe de champ de vecteurs holomorphe semi-complet et nous demontrons que c'est une condition necessaire pour qu'un germe de champ holomorphe puisse etre realise par un champ holomorphe complet defini sur une variete complexe. Comme premiere application de la notion de champ semi-complet, nous demontrons qu'en dimension complexe deux, le deuxieme jet d'un champ holomorphe complet, defini sur une surface complexe, en une singularite isolee ne s'annule pas. Nous donnons aussi un resultat analogue a celui ci-dessus, bien que moins puissant, valable cependant en toutes les dimensions. Dans la deuxieme partie de ce travail, nous considerons les singularites isolees des champs holomorphes complets, definis sur une certaine surface complexe, ou le premier jet du champ s'annule. Nous decrivons, a conjugaison holomorphe pres, tous les modeles locaux de ces champs. Finalement nous donnons des applications de ces resultats aux surfaces complexes compactes, surfaces elliptiques et aux germes des champs bi-dimensionnels semi-complets dont la partie lineaire est nilpotente

It is a remarkable fact that among the known examples of quadratic semicomplete vector fields on C^3, it is always possible to find linear coordinates where the corresponding vector field has all-or "almost all"-coefficients in the real numbers. Indeed, the coefficients are very often integral. The space of quadratic vector fields on C^3, up to linear equivalence, is a complex 9-dimensional family. The main result of this thesis establishes that the degree of freedom in determining the coefficients of a semicomplete vector field (under very mild generic assumptions) is at most 3. In other words, there are 3 parameters from which all remaining parameters are determined. Moreover if these 3 parameters are real, then so is the vector field. We start by considering a generic quadratic vector field Z on C^n that is homogeneous and is not a multiple of the radial vector field. The first step in our work will be to construct a canonical form for the induced vector field X on CP(n-1). This canonical form will be invariant under the action of a specific group of symmetries. When n=3, we then push further our approach by studying the singularities not lying on the exceptional divisor but at the hyperplane at infinity Delta=CP(2). In this setting the dynamics of the foliation turn out to be quite simple while the singularities tend to be degenerated. The advantage is that we can deal with degenerated singularities with the technique of successive blow-ups. This leads to simple expressions for the eigenvalues directly in terms of the coefficients of X.; Il est vraiment remarquable le fait que parmi les exemples connus de champs de vecteurs quadratiques semicomplets, il est toujours possible de trouver des coordonnées linéaires où le champ de vecteurs correspondant a tous -ou "presque tous"- ses coefficients dans l'ensemble des nombres réels. En effet, les coefficients sont très souvent entiers. L'espace des champs quadratiques en C^3, à équivalence linéaire près, est une famille de dimension complexe 9. Le résultat principal de cette thèse établi que les degrés de liberté pour déterminer les coefficients d'un champ de vecteurs semi-complet (sous des hypothèses génériques très faibles) est au plus 3. Autrement dit, il y a 3 paramètres à partir desquels tous les autres coefficients peuvent être obtenus dans un sens naturel. En particulier, si ces 3 coefficients sont réels, alors tous les coefficients sont réels. Nous commençons par considérer un champ quadratique générique Z en C^n, homogène et qui n'est pas un multiple du champ de vecteurs radial. Le premier pas dans notre travail sera de construire une forme canonique pour le champ de vecteurs X induit sur CP(n-1); Cette forme canonique est invariante sous l'action d'un groupe particulier de symétries. Lorsque n=3, nous pouvons améliorer notre approche en étudiant les singularités non pas sur le diviseur exceptionnel mais sur l'hyperplan à l'infini Delta = CP(2). Dans ce contexte la dynamique du feuilletage devient assez simple alors que les singularités ont tendance à devenir dégénérées. L'avantage est que l'on peut travailler avec des singularités dégénérées avec la technique des éclatements successifs. Ceci aboutit a des expressions simples pour les valeurs propres directement en terme des coefficients de X.

It is a remarkable fact that among the known examples of quadratic semicomplete vector fields on C^3, it is always possible to find linear coordinates where the corresponding vector field has all-or "almost all"-coefficients in the real numbers. Indeed, the coefficients are very often integral. The space of quadratic vector fields on C^3, up to linear equivalence, is a complex 9-dimensional family. The main result of this thesis establishes that the degree of freedom in determining the coefficients of a semicomplete vector field (under very mild generic assumptions) is at most 3. In other words, there are 3 parameters from which all remaining parameters are determined. Moreover if these 3 parameters are real, then so is the vector field. We start by considering a generic quadratic vector field Z on C^n that is homogeneous and is not a multiple of the radial vector field. The first step in our work will be to construct a canonical form for the induced vector field X on CP(n-1). This canonical form will be invariant under the action of a specific group of symmetries. When n=3, we then push further our approach by studying the singularities not lying on the exceptional divisor but at the hyperplane at infinity Delta=CP(2). In this setting the dynamics of the foliation turn out to be quite simple while the singularities tend to be degenerated. The advantage is that we can deal with degenerated singularities with the technique of successive blow-ups. This leads to simple expressions for the eigenvalues directly in terms of the coefficients of X.; Il est vraiment remarquable le fait que parmi les exemples connus de champs de vecteurs quadratiques semicomplets, il est toujours possible de trouver des coordonnées linéaires où le champ de vecteurs correspondant a tous -ou "presque tous"- ses coefficients dans l'ensemble des nombres réels. En effet, les coefficients sont très souvent entiers. L'espace des champs quadratiques en C^3, à équivalence linéaire près, est une famille de dimension complexe 9. Le résultat principal de cette thèse établi que les degrés de liberté pour déterminer les coefficients d'un champ de vecteurs semi-complet (sous des hypothèses génériques très faibles) est au plus 3. Autrement dit, il y a 3 paramètres à partir desquels tous les autres coefficients peuvent être obtenus dans un sens naturel. En particulier, si ces 3 coefficients sont réels, alors tous les coefficients sont réels. Nous commençons par considérer un champ quadratique générique Z en C^n, homogène et qui n'est pas un multiple du champ de vecteurs radial. Le premier pas dans notre travail sera de construire une forme canonique pour le champ de vecteurs X induit sur CP(n-1); Cette forme canonique est invariante sous l'action d'un groupe particulier de symétries. Lorsque n=3, nous pouvons améliorer notre approche en étudiant les singularités non pas sur le diviseur exceptionnel mais sur l'hyperplan à l'infini Delta = CP(2). Dans ce contexte la dynamique du feuilletage devient assez simple alors que les singularités ont tendance à devenir dégénérées. L'avantage est que l'on peut travailler avec des singularités dégénérées avec la technique des éclatements successifs. Ceci aboutit a des expressions simples pour les valeurs propres directement en terme des coefficients de X.