Jonny Ardila Ardila
We study the existence of first integral for holomorphic foliations in different scenarios and under different conditions, for instance germ of foliations given by vector fields and having a formal first integral or infinitely many invariant hypersurfaces.
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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.
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.
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.
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.
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 meager subset of the manifold.
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 study a special kind of local invariant sets of singular holomorphic foliations called nodal separators. We define notions of equisingularity and topological equivalence for nodal separators as intrinsic objects and, in analogy with the celebrated theorem of Zariski for analytic curves, we prove the equivalence of these notions. We give some applications in the study of topological equivalences of holomorphic foliations. In particular, we show that the nodal singularities and its eigenvalues in the resolution of a generalized curve are topological invariants.
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.
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 $p$.
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.
We prove that any holomorphic codimension 1 foliation on the complex projective plane has at most one singular point up to the action of an ad-hoc birational self map of the complex projective plane into itself. Consequently, any algebraic foliation on the affine plane has no singularities up to the action of a suitable birational self map of the complex projective plane into itself.
The Separatrix Theorem of C. Camacho and P. Sad guarantees the existence of invariant curve (separatrix) passing through the singularity of germ of holomorphic foliation on complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. Under some assumptions, we obtain existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. It will be necessary to require an extra condition of the foliation, namely, absence of saddle-node in its reduction of singularitie…
In this paper we study the dynamics of a holomorphic vector field near a singular point in dimension two using asymptotic expansion techniques. We consider a holomorphic vector field admitting first integrals in small sectors with nonzero asymptotic expansion. Under some natural hypotheses we prove the existence of a holomorphic first integral.
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。
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…
We consider a singular holomorphic foliation $\uF$ defined near a compact curve $\uC$ of a complex surface. Under some hypothesis on $(\uF,\uC)$ we prove that there exists a system of tubular neighborhoods $U$ of a curve $\underline{\mc D}$ containing $\uC$ such that every leaf $L$ of $\uF_{|(U\setminus \underline{\mc D})}$ is incompressible in $U\setminus \underline{\mc D}$. We also construct a representation of the fundamental group of the complementary of $\underline{\mc D}$ into a suitable automorphism group, which allows to state the topological classification of the germ of $(\uF,\uD)$, …
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.
We study those real $\mathcal{C}^\infty$ foliations in complex surfaces whose leaves are holomorphic curves. The main motivation is to try and understand these foliations in neighborhoods of curves: can we expect the space of foliations in a fixed neighborhood to be infinite-dimensional, or are there some contexts under which every such foliation is holomorphic? We give some restrictions and study in more details the geometry of foliations whose leaves belong to a holomorphic family of holomorphic curves. In particular, we classify all real-analytic foliations on neighborhoods of curves which …
In this work, we study dominant rational maps preserving singular holomorphic codimension one foliations on projective manifolds and that exhibit non-trivial transverse dynamics.
We present a list of open questions in the theory of holomorphic foliations, possibly with singularities. Some problems have been around for a while, others are very accessible.
We prove that a Morse type codimension one holomorphic foliation is not transverse to a sphere in the complex affine space. Also we characterize the variety of contacts of a linear foliation with concentric spheres.
We study Segre varieties associated to Levi-flat subsets in complex manifolds and apply them to establish local and global results on the integration of tangent holomorphic foliations.
We study codimension one holomorphic foliations on complex projective spaces and compact manifolds under the assumption that the foliation has a projective transverse structure in the complement of some invariant codimension one analytic subset. The basic motivation is the characterization of pull-backs of Riccati foliations on projective spaces. Our techniques apply to give a description of the generic models of codimension one foliations on compact manifolds of dimension $\ge 3$.
These are slightly informal lecture notes intended for graduate students about the standard local theory of holomorphic foliations and vector fields. Though the material presented here is well-known some of the proofs differs slightly from the classical arguments.
We describe a general construction providing index theorems localizing the Chern classes of the normal bundle of a subvariety inside a complex manifold. As particular instances of our construction we recover both Lehmann-Suwa's generalization of the classical Camacho-Sad index theorem for holomorphic foliations and our index theorem for holomorphic maps with positive dimensional fixed point set. Furthermore, we also obtain generalizations of recent index theorems of Camacho-Movasati-Sad and Camacho-Lehmann for holomorphic foliations transversal to a subvariety.
In this paper it is shown that the existence of two independent holomorphic first integrals for foliations by curves on (C^3,0) is not a topological invariant. More precisely, we provide an example of two topologically equivalent foliations such that only one of them admits two independent holomorphic first integrals. The existence of invariant surfaces over which the induced foliation possesses infinitely many separatrices possibly constitutes the sole obstruction for the topological invariance of complete integrability and a characterization of foliations admitting this type of invariant sur…
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.
The purpose of this paper is the study of vanishing cycles in holomorphic foliations by complex curves on compact complex manifolds. The main result consists in showing that a vanishing cycle comes together with a much richer complex geometric object - we call this object a foliated shell.
We prove that a one-dimensional foliation with generic singularities on a projective space, exhibiting a Lie group transverse structure in the complement of some codimension one algebraic subset is logarithmic, i.e., it is the intersection of codimension one foliations given by closed one-forms with simple poles. If there is only one singularity in a suitable affine space, then the foliation is induced by a linear diagonal vector field.
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 show that the set of singular holomorphic foliations of the projective spaces with split tangent sheaf and with good singular set is open in the space of holomorphic foliations. As applications we present a generalization of a result by Camacho-Lins Neto about linear pull-back foliations, we give a criterium for the rigidity of $\mathcal L$-foliations of codimension $k \ge 2$ and prove a conjecture by Cerveau-Deserti about the rigidity of a codimension one $\mathcal L$-foliation of $\mathbb P^4$. These results allow us to exhibit some previously unknown irreducible components of the spaces …
Heat diffusions on holomorphic foliations with non-degenerate singularities
本文研究紧复流形上带非退化奇点的 Brody 双曲全纯叶状结构，证明叶状热扩散与抽象热扩散一致，并由此得出抽象热扩散的唯一性。
Poincaré metric of holomorphic foliations with non-degenerate singularities
本文研究紧复流形上具有非退化奇点的布罗迪双曲叶层，证明其叶向庞加莱度量在奇点集附近具有带对数斜率的横向赫尔德连续性。
Finiteness of hyperbolic entropy for holomorphic foliations with non-degenerate singularities
本文研究了具有非退化奇点的全纯叶系的超几何熵有限性问题，证明其熵值为有限。
In dimensions greater than or equal to 3, the local structure of a singular holomorphic foliation conceals a globally defined foliation on the projective space of dimension one less. In this paper, we will discuss how the global dynamics of the latter foliation exerts influence on several problems that apparently have a purely local nature. In the course of the discussion, a few recent results and open problems in the area will be reviewed as well.
This manuscript is an introduction to the theory of holomorphic foliations on the complex projective plane. Historically the subject has emerged from the theory of ODEs in the complex domain and various attempts to solve Hilbert's 16th Problem, but with the introduction of complex algebraic geometry, foliation theory and dynamical systems, it has now become an interesting subject of its own.
Reeb flows, pseudo-holomorphic curves and transverse foliations
本文是一篇短篇综述，介绍了基于辛化空间伪全纯曲线理论的Reeb动力学最新研究进展，重点聚焦临界能量曲面附近的横截叶状结构，梳理了该理论对哈密顿动力学及天体力学经典问题的推动作用。
We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. More general, we prove that all nonsingular codimension one distributions on intermediary or generic Hopf manifolds are integrable and has holomorphic integral first. Also, we prove some results about singular holomorphic distributions on Hopf manifolds.
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…
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 $D$.
Let $\mathcal F$ be a holomorphic foliation on a compact Káhler surface with hyperbolic singularities and no foliation cycle. We prove that if the limit set of $\mathcal F$ has zero Lebesgue measure, then its complement is a modification of a Stein domain. The proof consists in building, in several steps, a metric of positive curvature for the normal bundle of $\mathcal F$ near the limit set. Then we construct a proper strictly plurisubharmonic exhaustion function for the complement of the limit set, by adapting Brunella's ideas to our singular context. The arguments hold more generally when t…
In this work, we begin by showing that a holomorphic foliation with singularities is reduced if and only if its normal sheaf is torsion free. In addition, when the codimension of the singular locus is at least two, it is shown that being reduced is equivalent to the reflexivity of the tangent sheaf. Our main results state on one hand, that the tangent sheaf of a codimension one foliation in P^3 is locally free if and only the singular scheme is a curve, and that it splits if and only if that curve is arithmetically Cohen-Macaulay. On the other hand, we discuss when a split foliation in P^3 is …
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…
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 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.
In this paper we study holomorphic foliations on $\mathbb{P}^2$ with only one singular point. If the singularity has algebraic multiplicity one, we prove that the foliation has no invariant algebraic curve. We also present several examples of such foliations in degree three.
Unique ergodicity for singular holomorphic foliations of $\mathbb{P}^3(\mathbb{C})$ with an invariant plane
本文将平面叶层的唯一遍历性结果推广到带无叶层循环不变平面的三维射影空间奇异全纯叶层，通过改造 Deroin-Kleptsyn 动力学方法并利用 Nguyên 可积性估计完成证明。
We study holomorphic foliations of codimension $k\geq 1$ on a complex manifold $X$ of dimension $n+k$ from the point of view of the exceptional minimal set conjecture. For $n\geq 2$ we show in particular that if the holomorphic normal bundle $N_{\mathcal{F}}$ is Griffiths positive, then the foliation does not admit a compact invariant set that is a complete intersection of $k$ smooth real hypersurfaces in $X$.
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.
On the holomorphic foliations admitting a common invariant algebraic set
本文研究全纯叶状结构具有公共不变代数集的条件，刻画了相关向量场模可自由生成的若干情形，并在二维弱驯顺多项式情形下给出模由两个向量场自由生成的充要条件。
Let \Fc be a holomorphic foliation by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: (1) the singular points of \Fc are all hyperbolic; (2) \Fc is Brody hyperbolic. Then we establish cohomological formulas for the Lyapunov exponent and the Poincaré mass of an extremal positive \ddc-closed current tangent to \Fc. If, moreover, there is no nonzero positive closed current tangent to \Fc, then we show that the Lyapunov exponent χ(\Fc) of \Fc, which is, by definition, the Lyapunov exponent of the unique normalized positive \ddc-closed cur…
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 $\leq n/k.$
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。
This paper is devoted to the resolution of singularities of holomorphic vector fields and of one-dimensional holomorphic foliations in dimension 3 and it has two main objectives. First, from the general perspective of one-dimensional foliations, we build upon the work of Cano-Roche-Spivakovsky and essentially complete it. As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan-Panazzolo but proved by means of rather different methods. The second objective of this paper consists of looking at a special class of singularities of foliations conta…
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…
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…
We prove inequalities relating the degrees of holomorphic distributions and of holomorphic foliations forming a flag on $\mathbb{P}^n$. Such inequalities are inspired by the so called Poincaré problem for foliations.
We study the holonomy cocycle H of a holomorphic foliation \Fc by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: 1) its singularities E are all hyperbolic; 2) there is no holomorphic non-constant map \C\to X such that out of E the image of \C is locally contained in a leaf. Let T be a harmonic current tangent to \Fc which does not give mass to any invariant analytic curve. Using the leafwise Poincaré metric, we show that H is integrable with respect to T. Consequently, we infer the existence of the Lyapunov exponent function of T.
Hyperbolic entropy for harmonic measures on singular holomorphic foliations
本文研究了奇异全纯叶系上调和测度的双曲熵性质，证明其在几乎处处为叶状常数，并展示了熵下界为2。
Nilpotent Holomorphic Foliations on $(\mathbb{C}^n,\mathbf{0})$
本文研究复数维$(\mathbb{C}^n,\mathbf{0})$上的nilpotent光滑叶结构，通过分析其与Takens正常型的关系，提出新的准则以判断叶结构是否为广义超曲面类型。
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