Y. Genzmer
We study some kind of rigidity property for dicritical foliation in the complex plane. In fact, we prove that for a generic dicritical foliation, there exists deformations of the resolution space which cannot carry any deformation of the foliation with constant holonomy pseudo-group. This situation never occurs in the non-dicritical case.
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We show the existence of an essentially unique logarithmic model for any germ of non-dicritical singular holomorphic foliation of codimension one in $({\mathbb C}^n,0)$ without saddle-nodes.
We study in this paper several properties concerning singularities of foliations in $(\mathbb{C}^3,\mathbf{0})$ that are pull-back of dicritical foliations in $(\mathbb{C}^2,\mathbf{0})$. Particularly, we will investigate the existence of first integrals (holomorphic and meromorphic) and the dicriticalness of such a foliation. In the study of meromorphic first integrals we follow the same method used by R. Meziani and P. Sad in dimension two. While the foliations we study are pull-back of foliations in $(\mathbb{C}^2,\mathbf{0})$, the adaptations are not straightforward.
By introducing a new invariant called the set of slidings, we give a complete strict classification of the class of germs of non-dicritical holomorphic foliations in the plan whose Camacho-Sad indices are not rational. Moreover, we will show that, in this class, the new invariant is finitely determined. Consequently, the finite determination of the class of isoholonomic non-dicritical foliations and absolutely dicritical foliations that have the same Dulac maps are proved.
We solve the Poincaré problem for plane foliations with only one dicritical divisor. Moreover, in this case, we give an algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case. We also provide an algorithm to compute a rational first integral of prefixed genus $g\neq 1$ of any type of plane foliation $\cf$. When the number of dicritical divisors dic$(\cf)$ is larger than two, this algorithm depends on suitable families of invariant curves. When dic$(\cf) = 2$, it proves that the degree of the rational first integral can be bounded only i…
We give a classification of absolutely dicritical foliations of cusp type, that is, the germ of singularities of complex foliations in the complex plane topologically equivalent to the singularity given by the level of the meromorphic function \frac{y^{2}+x^{3}}{xy}.
Curves of tangencies of foliation pairs and normalizing transformations
本文研究了由解析微分方程产生的叶状映射对的切线曲线集合，通过局部模型和正则化变换给出其描述，并证明在一般性假设下，任何具有相交光滑分支的解析曲线均可作为叶状映射对的切线曲线。
We develop a study on local polar invariants of planar complex analytic foliations at $(\mathbb{C}^{2},0)$, which leads to the characterization of second type foliations and of generalized curve foliations, as well as a description of the $GSV$-index. We apply it to the Poincaré problem for foliations on the complex projective plane $\mathbb{P}^{2}_{\mathbb{C}}$, establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve $S$ in terms of the degree of the foliation $\mathcal{F}$. We characterize the existence of a solution for th…
In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliations in dimension two and their invariant curves. It appears a characterization of second type foliations and generalized curve foliations as well as a description of the GSV-index in terms of polar curves. We also interpret the proofs concerning the Poincaré problem with polar invariants.
In this paper we give complete analytic invariants for germs of holomorphic foliations in $(\mathbb{C}^2,0)$ that become regular after a single blow-up. Some of them describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in finite dimensional complex vector space. Such singularities admit separatrices tangent to any direction at the origin. When enough separatrices coincide with their tangent directions (a condition that can always be attained if the mutiplicity of the germ at the origin is at most four) we are able to describe and realize …
In this paper we study bilipschitz equivalences of germs of holomorphic foliations in $(\mathbb{C}^2,0)$. We prove that the algebraic multiplicity of a singularity is invariant by such equivalences. Moreover, for a large class of singularities, we show that the projective holonomy representation is also a bilipschitz invariant.
We give an algorithm to decide whether an algebraic plane foliation F has a rational first integral and to compute it in the affirmative case. The algorithm runs whenever we assume the polyhedrality of the cone of curves of the surface obtained after blowing-up the set B_F of infinitely near points needed to get the dicritical exceptional divisors of a minimal resolution of the singularities of F. This condition can be detected in several ways, one of them from the proximity relations in B_F and, as a particular case, it holds when the cardinality of B_F is less than 9.
Let F be a germ of a singular foliation of the complex plane. Assuming that F is a generalized curve D. Marin and J.-F. Mattei proved the incompressibility of the foliation in a neighborhood from which a finite set of analytic curves is removed. We show in the present work that this hypothesis cannot be eluded by building examples of foliations, reduced after one blow-up, for which the property does not hold. Even if we manage to prove that the individual saddle-node foliation is incompressible, their leaves not retracting tangentially on the boundary of the domain of definition forbids a gene…
Let $\mathcal{F}$ be a singular codimension one holomorphic foliation on a compact complex manifold $X$ of dimension at least three such that its singular set has codimension at least two. In this paper, we determine Lehmann-Suwa residues of $\mathcal{F}$ as multiples of complex numbers by integration currents along irreducible complex subvarieties of $X$. We then prove a formula that determines the Baum-Bott residue of simple almost Liouvillian foliations of codimension one, in terms of Lehmann-Suwa residues, generalizing a result of Marco Brunella. As an application, we give sufficient condi…
Analytic semiroots for plane branches and singular foliations
本文研究等奇异平面分支的解析模，提出通过扩充标准基定义解析半根，并证明增广标准基在几何与叶状结构下具有良好性质，其临界包中分支的微分值半模可由初始半模生成元列表的截断描述。
We consider the problem of extending germs of plane holomorphic foliations to foliations of compact surfaces. We show that the germs that become regular after a single blow up and admit meromorphic first integrals can be extended, after local changes of coordinates, to foliations of compact surfaces. We also show that the simplest elements in this class can be defined by polynomial equations. On the other hand we prove that, in the absence of meromorphic first integrals there are uncountably many elements without polynomial representations.
We build a germ of singular foliation in dimension two with analytical class of separatrix and holonomy representations prescribed. Thanks to this construction, we study the link between moduli of a foliation and moduli of its separatrix.
We give geometric and algorithmic criterions in order to have of a proper Galois closure for a codimension one germ of quasi-homogeneous foliation. We recall this notion recently introduced by B. Malgrange, and describe the Galois envelope of a group of germs of analytic diffeomorphisms. The geometric criterions are obtained from transverse analytic invariants, whereas the algorithmic ones make use of formal normal forms.
This paper deals with the question of analytic continuation of holonomy germs of holomorphic foliations. We prove that for a quasi-minimal Riccati foliation of the complex projective plane, any holonomy germ of the foliation between complex projective lines can be analytically continued along a generic Brownian path.
The object of this survey is to give an overview on the topology of singularities of holomorphic foliation germs on $(\mathbb C^2,0)$.
We prove that a $C^{\infty}$ equivalence between germs holomorphic foliations at $({\mathbb C}^2,0)$ establishes a bijection between the sets of formal separatrices preserving equisingularity classes. As a consequence, if one of the foliations is of second type, so is the other and they are equisingular.
We consider germs of holomorphic vector fields with an isolated singularity at the origin $0\in\mathbb{C}^2$. We introduce a notion of stability, similar to "Lyapunov stability". For such a germ, called $L$-stable singularity, either the corresponding foliation admits a holomorphic first integral, or it is a real logarithmic foliation singularity. A notion of $L$-stability is also naturally introduced for a leaf of a foliation. In the complex codimension one case, for holomorphic foliations, the holonomy groups of $L$-stable leaves are proved to be abelian, of a suitable type. This implies the…
In this paper we study the problem of analytic extension of germs of holonomy 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 germs of holonomy between a fiber and a 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 are included and dense in the limit set and uncountable, giving another negative answer to a…
We prove a result of classification for germs of formal and convergent quasi-homogeneous foliations in C^2 with fixed separatrix. Basically, we prove that the analytical and formal class of such a foliation depend respectively only on the analytical and formal class of its representation of projective holonomy.
We investigate residue-type indices for germs of holomorphic foliations in the plane and characterize second type foliations - those not containing tangent saddle-nodes in the reduction of singularities - by an expression involving the Baum-Bott, variation and polar excess indices. These local results are applied in the study of logarithmic foliations on compact complex surfaces.
In this paper, we will construct a pre-normal form for germs of codimension one holomorphic foliation having a particular type of separatrix, of cuspidal type. We will also give a sufficient condition, in the quasi-homogeneous, three-dimensional case, for these foliations to be of generalized surface type.
We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy contains all the relevant dynamical information, in particular the projective holonomy representation whose topological invariance was conjectured in the eighties by Cerveau and Sad and proved here under mild hypotheses.
This work deals with the topological classification of germs of singular foliations on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we compute the moduli space of topological classes in terms of the cohomology of a new algebraic object that we call group-graph. This moduli space may be an infinite dimensional functional space but under generic conditions we prove that it has finite dimension and we describe its algebraic and topological struct…
This work deals with the topological classification of singular foliation germs on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we prove the existence of a topological universal deformation through which every equisingular deformation uniquely factorizes up to topological conjugacy. This is done by representing the functor of topological classes of equisingular deformations of a fixed foliation. We also describe the functorial dependence of th…
Holonomy and equivalence of analytic foliations
本文研究一类由特定向量场芽生成的奇异叶状结构，证明在适当假设下，其解析分类可归结为沿给定分界线的局部和乐的解析共轭。
Topological Moduli Space for Germs of Holomorphic Foliations III: Complete families
本文利用一般奇异叶层芽的拓扑分类结果，构造了包含所有拓扑类的极小完全族，并证明任意等奇异全局族均可通过该族分解。
Stability and Logarithmic Foliations on Complex Projective Spaces
本文在复射影空间上定义射影$L$-稳定性，证明其对极大不变代数除子与非临界非节点奇点条件可推出全局对数叶状结构，并应用于$\mathbb P^2$的拓扑刚性。
Stability of closed leaves holomorphic foliations via torsion behavior of groups of germs
本文通过引入胚群扭转轨迹，证明了闭叶全纯叶状结构中存在高维紧致叶不变解析集，且叶体积一致有界，并给出射影情形的全局化。
A singular real analytic foliation $\mathcal{F}$ of real codimension one on an $n$-dimensional complex manifold $M$ is Levi-flat if each of its leaves is foliated by immersed complex manifolds of dimension $n-1$. These complex manifolds are leaves of a singular real analytic foliation $\mathcal{L}$ which is tangent to $\mathcal{F}$. In this article, we classify germs of Levi-flat foliations at $(\mathbb{C}^{n},0)$ under the hypothesis that $\mathcal{L}$ is a germ holomorphic foliation. Essentially, we prove that there are two possibilities for $\mathcal{L}$, from which the classification of $\…
We are interested in characterizing the holonomy maps associated to integral curves of non-degenerate singularities of holomorphic vector fields. Such a description is well-known in dimension 2 where is a key ingredient in the study of reduced singularities. The most intricate case in the 2 dimensional setting corresponds to (Siegel) saddle singularities. This work treats the analogous problem for saddles in higher dimension. We show that any germ of holomorphic biholomorphism, in any dimension, can be obtained as the holonomy map associated to an integral curve of a saddle singularity. A natu…
On neighborhoods of embedded toroidal and Hopf manifolds and their foliations
本文研究嵌入复流形的邻域芽全纯等价于其法丛零截面邻域芽的问题，给出两类全新例子及全纯叶状结构存在条件，丰富了复几何相关研究。
We study complex Lie algebras spanned by pairs \left(Z,Y\right) of germs of a meromorphic vector field of the complex plane satisfying \left[Z,Y\right]=δY for some δ\in\ww C . This topic relates to Liouville-integrability of the differential equation induced by the foliation underlying Z . We give a direct geometric proof of a result by M. Berthier and F. Touzet characterizing germs of a foliation admitting a first-integral in a Liouvillian extension of the standard differential field. In so doing we study transverse and tangential rigidity properties when Z is holomorphic and its linear part …
On flags of holomorphic foliations associated with singular second-order ordinary differential equations
本文研究与二阶常微分方程相关的全纯叶状结构2-旗，分析其奇点集性质，给出容许该结构的二阶方程分类，提出高维2-旗芽构造方法，并证明二阶及以上方程形式上均对应$\boldsymbol{\rm \bf C}^3$原点处的2-旗芽。
We study groups of germs of complex diffeomorphisms having a property called irreducibility. The notion is motivated by a similar property of the fundamental group of the complement of an irreducible hypersurface in the complex projective space. Natural examples of such groups of germ maps are given by holonomy groups and monodromy groups of integrable systems (foliations) under certain conditions on the singular or ramification set. The case of complex dimension one is studied in [7] where finiteness is proved for irreducible groups under certain arithmetic hypothesis on the linear part. In d…
This article studies germs of holomorphic vector fields at the origin of C3 that are tangent to holomorphic foliations of codimension one. Two situations are considered. First, we assume hypotheses on the reduction of singularities of the vector field - for instance, that the final models belong to a family of vector fields whose linear parts have eigenvalues satisfying a condition of non-resonance - in order to conclude that the foliation is of complex hyperbolic type, that is, without saddle-nodes in its reduction of singularities. In the second part, we prove that a vector field that is tan…
We consider a non-dicritic germ of foliations defined in some ball, with finite number of separatrices and satisfying some additional but generic hypothesis. We prove that there exists an open neighborhood U of the total separatice set S such that each leaf of foliaion restrict to (U-S) is incompressible in (U-S).
Holomorphic foliations tangent to Rolle-pfaffian hypersurfaces
本文研究复平面原点处满足Rolle-Khovanskii条件的与Pfaffian超曲面相切的全纯叶状结构芽，证明其由闭亚纯1-形式定义，并完成奇点消解中简单模型的分类。
We study under the standpoint of integrable complex analytic 1-forms (complex analytic foliations), a class of second order ordinary differential equations with periodic coefficients. More precisely, we study Hill's equations of motion of the moon, which are related to the dynamics of the system Sun-Earth-Moon. We associate to the {\em complex Hill equation} an integrable complex analytic one-form in dimension three. This defines a {\it Hill foliation}. The existence of first integral for a Hill foliation is then studied. The simple cases correspond to the existence of rational or Liouvillian …
We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. Each element f in this class induces a germ of foliation (df = 0). Proceeding similarly to the homogeneous case and the quasi homogeneous case treated by Genzmer and Paul, we describe the local moduli space of the foliations in this class and give analytic normal forms. We prove also the uniqueness of these normal forms.
On germs of constriction curves in model of overdamped Josephson junction, dynamical isomonodromic foliation and Painlevé 3 equation
本文研究过阻尼约瑟夫森结模型中旋转数水平集的收缩曲线结构，证明其极限点由贝塞尔函数零点给出，并证明由 Painlevé 3 方程控制的动态等单导叶状结构的 Poincaré 映射在相应区域良定义且将相邻极限点依次对应。
We study analytic deformations and unfoldings of holomorphic foliations in complex projective plane $\mathbb{C}P(2)$. Let $\{\mathcal{F}_t\}_{t \in \mathbb{D}_ε}$ be topological trivial (in $\mathbb{C}^2$) analytic deformation of a foliation $\mathcal{F}_0$ on $\mathbb{C}^2$. We show that under some dynamical restriction on $\mathcal{F}_0$, we have two possibilities: $\mathcal{F}_0$ is a Darboux (logarithmic) foliation, or $\{\mathcal{F}_t\}_{t \in \mathbb{D}_ε}$ is an unfolding. We obtain in this way a link between the analytical classification of the unfolding and the one of its germs at the…
An algebraizable singularity is a germ of a singular holomorphic foliation which can be defined in some appropriate local chart by a differential equation with algebraic coefficients. We show that there exists at least countably many saddle-node singularities of the complex plane that are not algebraizable.
Rigidity of saddle loops
本文研究了鞍点环的刚性性质，证明其通过庞加莱映射分类，并在多值映射情况下展现形式刚性，同时列举了所有可积鞍点环的解析类。
Here we state a conjecture concerning a local version of Brunella's alternative: any codimension one foliation in $({\mathbb C}^3,0)$ without germ of invariant surface has a neighborhood of the origin formed by leaves containing a germ of analytic curve at the origin. We prove the conjecture for the class of codimension one foliations whose reduction of singularities is obtained by blowing-up points and curves of equireduction and such that the final singularities are free of saddle-nodes. The concept of "partial separatrix" for a given reduction of singularities has a central role in our argu…
Logarithmic models and meromorphic functions in dimension two
本文研究二维空间中对数模型的构造及其在亚纯函数和向量场中的应用，通过分离子和Camacho-Sad指数构建具有特定奇点结构的亚纯函数。
The Poincaré problem for reducible curves
本文研究了复曲面中局部全纯叶状结构的庞加莱问题，基于不变曲线芽数据给出了叶状结构重数的精确下界，并证明分支孤立的不变曲线重数至多为叶状结构重数的两倍，最后将局部方法推广至复射影平面。
We study groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the notion of holonomy group of a leaf of a foliation. For dimension one, there is a well-established dictionary relating analytic/formal classification of the group, with its algebraic properties (finiteness, commutativity, solvability, ...). Such system of equivalences also characterizes the existence of suitable {\it integrating factors}, i.e., invariant vector fields and one-forms associated…
We study analytic integrable deformations of the germ of a holomorphic foliation given by $df=0$ at the origin $0 \in \mathbb C^n, n \geq 3$. We consider the case where $f$ is a germ of an irreducible and reduced holomorphic function. Our central hypotheses is that, {\em outside of a dimension $\leq n-3$ analytic subset $Y\subset X$, the analytic hypersurface $X_f : (f=0)$ has only normal crossings singularities}. We then prove that, as germs, such deformations also exhibit a holomorphic first integral, depending analytically on the parameter of the deformation. This applies to the study of in…
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