Analytical and formal classifications of quasi-homogeneous foliations in (C^2,0)

6次齐次凸叶状结构

Homogeneous Convex Foliations of degree 6

本文研究复射影平面上的齐次凸叶状结构，利用射影对偶性关联叶状结构与对偶平面$d$-网，分类了具三个径向奇点的齐次凸叶状结构并完成6次凸齐次叶状结构分类。

Codimension two holomorphic foliations

This paper is devoted to the study of codimension two holomorphic foliations and distributions. We prove the stability of complete intersection of codimension two distributions and foliations in the local case. Converserly we show the existence of codimension two foliations which are not contained in any codimension one foliation. We study problems related to the singular locus and we classify homogeneous foliations of small degree.

On codimension one foliations with prescribed cuspidal separatrix

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.

由 $\mathbb{P}^2$ 上凸与齐次叶状结构的乘积生成的网

Webs Generated by Products of convex and homogeneous Foliations on $\mathbb{P}^2$

本文研究射影平面上的平坦网，提出通过有限多个凸约化或凸齐次叶状结构与不变直线的乘积构造此类网的两种方法，并证明在这两种情况下其对偶网均为平坦的。

Invariant holomorphic foliations on Kobayashi hyperbolic homogeneous manifolds

Let $M$ be a Kobayashi hyperbolic homogenous manifold. Let $\mathcal F$ be a holomorphic foliation on $M$ invariant under a transitive group $G$ of biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a holomorphic $G$-equivariant submersion $π\colon M \to N$ onto a $G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in $\mathbb C^n$, then the fixed point set of $\mathcal Q$ is either empty or a connected complex submanifold of $D$.

Analytic classification of a class of cuspidal foliations in $({\mathbb C}^3,0)$

In this article we study the analytic classification of certain types of quasi-homogeneous cuspidal holomorphic foliations in $(\CC^3,{\bf 0})$ via the essential holonomy defined over one of the components of the exceptional divisor that appears in the reduction of the singularities of the foliation.

Absolutely $k$-convex domains and holomorphic foliations on homogeneous manifolds

We consider a holomorphic foliation $\mathcal{F}$ of codimension $k\geq 1$ on a homogeneous compact Kähler manifold $X$ of dimension $n>k$. Assuming that the singular set $Sing(\mathcal{F})$ of $\mathcal{F}$ is contained in an absolutely $k$-convex domain $U\subset X$, we prove that the determinant of normal bundle $\det(N_{\mathcal{F}})$ of $\mathcal{F}$ cannot be an ample line bundle, provided $[n/k]\geq 2k+3$. Here $[n/k]$ denotes the largest integer $\leq n/k.$

Differentiable equisingularity of holomorphic foliations

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.

Codimension one holomorphic foliations on $\mathbb P^n_{\mathbb C}$: problems in complex geometry

After a short review on foliations, we prove that a codimension 1 holomorphic foliation on $\mathbb P^3_{\mathbb C}$ with simple singularities is given by a closed rational 1-form. The proof uses Hironaka-Matsumura prolongation theorem of formal objects.

Homogeneous matchbox manifolds

We prove that a homogeneous matchbox manifold of any finite dimension is homeomorphic to a McCord solenoid, thereby proving a strong version of a conjecture of Fokkink and Oversteegen. The proof uses techniques from the theory of foliations that involve making important connections between homogeneity and equicontinuity. The results provide a framework for the study of equicontinuous minimal sets of foliations that have the structure of a matchbox manifold.

复平面上的一阶和四阶均匀预叶形及其Legendre变换定义的平坦四网

Homogeneous pre-foliations of co-degree one and degree four on the projective plane

本文通过射影自同构分类复平面上一阶和四阶均匀预叶形，其Legendre变换定义平坦四网，根据底层均匀叶形的类型区分不同情况，结合曲率-全纯性准则和符号计算得出有限列表。

几何正规化

Geometric normalization

针对平面无理椭圆不动点处的局部解析微分同胚，提出包含经典形式正规化的几何正规化概念，证明其对应唯一形式不变叶状结构，并得出几何正规化通有发散、即解析不变叶状结构通有不存在的结论。

Moduli spaces of a family of topologically non quasi-homogeneous functions

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.

Two dimensional neighborhoods of elliptic curves: formal classification and foliations

We classify two dimensional neighborhoods of an elliptic curve C with torsion normal bundle, up to formal equivalence. The proof makes use of the existence of a pair (indeed a pencil) of formal foliations having C as a common leaf, and the fact that neighborhoods are completely determined by the holonomy of such a pair. We also discussanalytic equivalence and show, for each formal model, that the corresponding moduli space is infinite dimensional.

Convex foliations of degree 5 on the complex projective plane

We show that up to automorphisms of $\mathbb P^2_{\mathbb C}$ there are $14$ homogeneous convex foliations of degree $5$ on $\mathbb P^2_{\mathbb C}.$ We establish some properties of the Fermat foliation $\mathcal F_{0}^{d}$ of degree $d\geq2$ and of the Hilbert modular foliation $\mathcal{F}_H^{5}$ of degree $5.$ As a consequence, we obtain that every reduced convex foliation of degree $5$ on $\mathbb P^2_{\mathbb C}$ is linearly conjugated to one of the two foliations $\mathcal F_{0}^{5}$ or $\mathcal{F}_H^{5},$ which is a partial answer to a question posed in $2013$ by D. Mar\'ın and J.V. P…

Tissus plats et feuilletages homogènes sur le plan projectif

The aim of this work is to study the foliations on the complex projective plane with flat \textsc{Legendre} transform (dual web). We establish some effective criteria for the flatness of the dual $d$-web of a homogeneous foliation of degree $d$ and we describe some explicit examples. These results allow us to show that up to automorphism of $\mathbb{P}^2$ there are $11$ homogeneous foliations of degree $3$ with flat dual web. We will see also that it is possible, under certain assumptions, to bring the study of flatness of the dual web of a general foliation to the homogeneous framework. We ge…

全纯叶状结构的广义Poincaré-Dulac奇点

Generalized Poincaré-Dulac singularities of holomorphic foliations

本文研究复二维全纯叶状结构中具有Poincaré-Dulac型约化的幂零奇点，通过例外除子分量的和乐给出解析分类，并证明此类奇点具有形式解析刚性。

Convex foliations of degree 4 on the complex projective plane

We show that up to automorphisms of $\mathbb{P}^2_{\mathbb C}$ there are $5$ homogeneous convex foliations of degree four on $\mathbb{P}^2_{\mathbb C}.$ Using this result, we give a partial answer to a question posed in $2013$ by D. {Marín} and J. {Pereira} about the classification of reduced convex foliations on~$\mathbb{P}^2_{\mathbb C}.$

Integrating factors for groups of formal complex diffeomorphisms

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…

Hirzebruch曲面上的孤立奇点叶结构

Foliations with isolated singularities on Hirzebruch surfaces

本文研究Hirzebruch曲面上叶结构的奇点性质，证明在特定条件下叶结构由其奇点方案决定，并探讨了不同δ值下的不同结论。

关于拟齐次曲线的分类

On the classification of quasi-homogeneous curves

本文应用全纯叶理技术描述 $(\mathbb{C}^2,0)$ 中拟齐次曲线芽的解析不变量，并提出一种有效方法判定两条拟齐次曲线是否解析等价。

Holonomy representation of quasi-projective leaves of codimension one foliations

We prove that a representation of the fundamental group of a quasi-projective manifold into the group of formal diffeomorphisms of one variable either is virtually abelian or, after taking the quotient by its center, factors through an orbicurve.

Groups of formal diffeomorphisms in several complex variables and closed one-forms

We study groups of formal diffeomorphisms in several complex variables. For abelian, metabelian or nilpotent groups we investigate the existence of suitable formal vector fields and closed differential forms which exhibit an invariance property under the group action. Our results are applicable in the construction of suitable integrating factors for holomorphic foliations with singularities. We believe they are a starting point in the study of the connection between Liouvillian integration and transverse structures of holomorphic foliations with singularities in the case of arbitrary codimensi…

光滑平面网曲率全纯性的一个判据及其在 $\mathbb{P}^{2}_{\mathbb{C}}$ 上齐次叶化对偶网中的应用

A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on $\mathbb{P}^{2}_{\mathbb{C}}$

本文建立了光滑平面网曲率沿判别式分支全纯的有效判据，并应用于刻画 $\mathbb{P}^{2}_{\mathbb{C}}$ 上齐次叶化对偶网的曲率全纯性与平坦性，完整分类了 Galois 循环与非循环情形。

Losik classes for codimension-one foliations

Following Losik's approach to Gelfand's formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for non-triviality in terms of dynamical properties of generators of the holonomy groups are found. The non-triviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class, e.g, the Mizutani-Morita-Tsuboi Theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. I…

关于与奇异二阶常微分方程相关的全纯叶状结构的旗（flags）

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。