Consider a field k of characteristic 0, not necessarily algebraically closed, and a fixed algebraic curve f=0defined by a tame polynomial f is an element of k[x,y] with only quasi-homogeneous singularities. We prove that the space of holomorphic foliations in the plane A(k)(2) having f=0 as a fixed invariant curve is generated as k[x,y]-module by at most four elements, three of them are the trivial foliations fdx,fdy and df. Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen-Suslin theorem, we show that for a suitable field extension K of k such a module over K[x,y] is actually generated by two elements, and therefore, such curves are free divisors in the sense of K. Saito. After performing Groebner basis for this module, we observe that in many well-known examples, K=k.

Consider a complex one-dimensional foliation on a complex surface near a singularity $p$. If ${\mathcal{I}}$ is a closed invariant set containing the singularity $p$, then ${\mathcal{I}}$ contains either a separatrix at $p$ or an invariant real three-dimensional manifold singular at $p$.

For a linear differential equation with a mild condition on its singularities, we discuss generalized continued fractions converging to expressions in its solutions and their derivatives. In the case of an order two linear differential equation, this is the logarithmic derivative of the holomorphic solution near a singularity.

We show that a germ of a holomorphic one-dimensional foliation at a singularity in a space of dimension two admits a holomorphic first integral if and only if there are infinitely many closed leaves and a finite number of separatrices, with each separatrix having linearizable holonomy. Indeed, if there are infinitely many closed leaves and the set of separatrices is finite, then the foliation admits either a holomorphic first integral or a formal simple integrating factor of Darboux type.

The present article deals with the classification of neighborhoods of negatively embedded submanifolds A of a complex manifold X. The main tools we use are one-dimensional foliations whose set of singularities is A and which are normally attracting at A. The linearization of these foliations is provided under general cohomological conditions. As a consequence, an extension of the classical embedding theorem of Grauert is obtained.

A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.

In this paper we will establish a structure theorem concerning the extension of analytic objects associated to germs of dimension one foliations on surfaces, through one-dimensional barriers. As an application, an extension theorem for projective transverse structures is obtained.

In this paper we present recent results concerning global aspects of $${\mathbb C}$$ and $${\mathbb C^*}$$ -actions on Stein surfaces. Our approach is based on a byproduct of techniques from Geometric Theory of Foliations (holonomy, stability), Potential theory (parabolic Riemann surfaces, Riemann-Koebe Uniformization theorem) and Several Complex Variables (Hartogs’ extension theorems, Theory of Stein spaces). Our main motivation comes from the original works of M. Suzuki and Orlik-Wagreich. Some of their results are extended to a more general framework. In particular, we prove some linearization theorems for holomorphic actions of $${\mathbb C}$$ and $${\mathbb C^*}$$ on normal Stein analytic spaces of dimension two. We also add a list of questions and open problems in the subject. The underlying idea is to present the state of the art of this research field.

In this paper we present recent results concerning global aspects of C and C*-actions on Stein surfaces. Our approach is based on a byproduct of techniques from Geometric Theory of Foliations (holonomy, stability), Potential theory (parabolic Riemann surfaces, Riemann-Koebe Uniformization theorem) and Several Complex Variables (Hartogs' extension theorems, Theory of Stein spaces). Our main motivation comes from the original works of M. Suzuki and Orlik-Wagreich. Some of their results are extended to a more general framework. In particular, we prove some linearization theorems for holomorphic actions of C and C* on normal Stein analytic spaces of dimension two. We also add a list of questions and open problems in the subject. The underlying idea is to present the state of the art of this research field.

In this article we study good ℂ* actions on Stein surfaces and we construct their moduli by means of the resolution data of the dicritical singularity of the action. We also classify ℂ* transversal actions around a Riemann surface embedded in a two dimensional manifold.

We prove that the limit of a sequence of generic semi-algebraic sets given by a finite number of formulas always exists and is a semi-algebraic set that can be explicitly given as a Boolean expression involving the primitives of the additive forms of the formulas.

We study and classify actions of the complex multiplicative group on a nonsingular Stein surface with an isolated nondicritical singularity. We prove that the corresponding foliation exhibits a holomorphic first integral of a type F = f n g m where f and g are global holomorphic functions and \({n, m \in \mathbb N}\). Under some additional conditions on the functions f and g we prove analytic linearization for the action. Our results can be viewed as extension of the original work of Masakazu Suzuki.

We study codimension one smooth foliations with Morse type singularities on closed manifolds. We obtain a description of the manifold if there are more centers than saddles. This result relies on and extends previous results of Reeb for foliations having only centers, results of Wagneur for foliations with Morse singularities and results of Eells and Kuiper for manifolds admitting Morse functions with three singularities.

In this paper we present recent results concerning global aspects of ℂ and ℂ^* -actions on Stein surfaces. Our approach is based on a byproduct of techniques from Geometric Theory of Foliations (holonomy, stability), Potential theory (parabolic Riemann surfaces, Riemann-Koebe Uniformization theorem) and Several Complex Variables (Hartogs’ extension theorems, Theory of Stein spaces). Our main motivation comes from the original works of M. Suzuki and Orlik-Wagreich. Some of their results are extended to a more general framework. In particular, we prove some linearization theorems for holomorphic actions of ℂ and ℂ^* on normal Stein analytic spaces of dimension two. We also add a list of questions and open problems in the subject. The underlying idea is to present the state of the art of this research field.

Let $V$ be an irreducible complex analytic space of dimension two with normal singularities and $\vr:\mathbb{C^*}\times V\to V$ a holomorphic action of the group $\mathbb{C^*}$ on $V$. Denote by $\fa_\vr$ the foliation on $V$ induced by $\vr$. The leaves of this foliation are the one-dimensional orbits of $\vr$. %and its singularities are the fixed points of $\vr$. We will assume that there exists a \emph{dicritical} singularity $p\in V$ for the $\bc^*$-action, i.e. for some neighborhood $p\in W\subset V$ there are infinitely many leaves of $\mathcal {F}_\vr|_{W}$ accumulating only at $p$. The closure of such a local leaf is an invariant local analytic curve called a \emph{separatrix} of $\mathcal{F}_\vr$ through $p$. In \cite{Orlik} Orlik and Wagreich studied the 2-dimensional affine algebraic varieties embedded in $\mathbb{C}^{n+1}$, with an isolated singularity at the origin, that are invariant by an effective action of the form $\sigma_Q(t,(z_{0},...,z_{n}))=(t^{q_{0}}z_{0},..., t^{q_{n}}z_{n})$ where $Q=(q_0,...,q_n) \in\mathbb N^{n+1}$, i.e. all $q_{i}$ are positive integers. Such actions are called \emph{good} actions. In particular they classified the algebraic surfaces embedded in $\mathbb{C}^{3}$ endowed with such an action. It is easy to see that any good action on a surface embedded in $\mathbb{C}^{n+1}$ has a dicritical singularity at $0\in\mathbb{C}^{n+1}$. Conversely, it is the purpose of this paper to show that good actions are the models for analytic $\mathbb{C^*}$-actions on Stein analytic spaces of dimension two with a dicritical singularity.

We study holomorphic flows on Stein manifolds. We prove that a holomorphic flow with isolated singularities and a dicritical singularity of the form \(\sum^{n}_{j=1}\lambda_{j}z_{j}\frac{\partial}{\partial z_{j}}+\ldots, \lambda_{j}\in \mathbb{Q}_{+},\forall j \in \{1,\ldots,n\}\) on a Stein manifold \(M^n, n \geq 2\) with \({\mathop{H}\limits^{\vee}}{^{2}}(M^{n}, {{{\mathbb{Z}}}})=0\), is globally analytically linearizable; in particular M is biholomorphic to \({\mathbb{C}}^{n}\). A complete stability result for periodic orbits is also obtained.

A closed, connected oriented three-manifold supporting a codimension one oriented smooth foliation with Morse singularities having more centers than saddles and without saddle connections is diffeomorphic to the three-sphere. The use of the Reeb Stability theorem in place of the Poincaré–Bendixson theorem paves the way to a three-dimensional version, for foliations with singularities of Morse type, of a classical result of Haefliger. Finally, we give an example of a codimension one C∞ foliation in the closed ball B¯4⊂R4, with only one singularity which is of saddle type 2–2 and transverse to the boundary S3=∂B4.