Percy Fernández, Liliana Puchuri, Rudy Rosas
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.
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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 $p$.
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…
After the nice result introduced by Belotto in [1] concerning the local monomialization of a singular foliation given by n first integrals, this work is a continuation in the same spirit. In this paper, we introduce a important conjecture about local monomialization of a singular foliation of Darboux type (see section 1). This conjecture can be used to study pseudo-abelian integrals [2,4].
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 show that for the C^1-open set of partially hyperbolic diffeomorphisms constructed in (M. Shub and A. Wilkinson, "Pathological foliations and removable zero exponents," Invent. math. 139 (2000) 3, 495-508), Lebesgue measure on the 3-torus decomposes as atomic measure along the leaves of the central foliation.
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.
Remarks on foliations on $\mathbb{CP}^2$ with a unique singular point
本文研究 $\mathbb{CP}^2$ 上具有唯一奇点的叶状结构，给出次数 7 的非临界情形下具有发散分隔线的显式例子，并探讨低次数下的不同族。
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.
A holomorphic pre-foliation $\mathscr{F}=\ell\boxtimes\mathcal{F}$ of co-degree $1$ and degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is the data of a line $\ell$ of $\mathbb{P}^{2}_{\mathbb{C}}$ and a holomorphic foliation $\mathcal{F}$ on $\mathbb{P }^{2}_{\mathbb{C}}$ of degree $d-1.$ We study pre-foliations of co-degree $1$ on $\mathbb{P}^{2}_{\mathbb{ C}}$ with a flat Legendre transform (dual web). After having established some general results on the flatness of the dual $d$-web of a homogeneous pre-foliation of co-degree $1$ and degree $d$, we describe some explicit examples and we show th…
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.
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。
This paper provides a canonical compactification of the plane ${\mathbb R}^2$ by adding a circle at infinity associated to a countable family of singular foliations or laminations (under some hypotheses), generalizing an idea by Mather \cite{Ma}. Moreover any homeomorphism of ${\mathbb R}^2 $ preserving the foliations extends on the circle at infinity. Then this paper provides conditions ensuring the minimality of the action on the circle at infinity induced by an action on ${\mathbb R}^2 $ preserving one foliation or two transverse foliations. In particular the action on the circle at infinit…
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 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.
Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for certain classes of singular hyperbolic flows in three dimensions. The results apply in particular to the classical Lorenz attractor. However, many of the proofs rely heavily on the smoothness of the stable foliation for the flow. In this paper, we show that many statistical properties hold for singular hyperbolic flows with no smoothness assumption on the stable foliation. These properties include existence of SRB measures, central limit theorems and associated …
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 $(\uF,\uD)$, …
Jacobian curve of singular foliations
该研究利用叶状结构的Camacho-Sad指标与共线点/除子概念刻画两奇异叶状结构的相似性，给出其雅可比曲线的分解方式，可复现平面曲线雅可比曲线、曲线及叶状结构极曲线的因式分解结果。
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.
The Cremona group of birational transformations of $\mathbb{P}^2_\mathbb{C}$ acts on the space $\mathbb{F}(2)$ of holomorphic foliations on the complex projective plane. Since this action is not compatible with the natural graduation of $\mathbb{F}(2)$ by the degree, its description is complicated. The fixed points of the action are essentially described by Cantat-Favre in \cite{CF}. In that paper we are interested in problems of "aberration of the degree" that is pairs $(ϕ,\mathcal{F})\in\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})\times\mathbb{F}(2)$ for which $\degϕ^*\mathcal{F}<(°\mathcal{F}+1)\d…
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 $\ge 3$.
This paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on projective manifolds.
Let F be a holomorphic foliation of P^2 by Riemann surfaces. Assume all the singular points of F are hyperbolic. If F has no algebraic leaf, then there is a unique positive harmonic $(1,1)$ current $T$ of mass one, directed by F. This implies strong ergodic properties for the foliation. We also study the harmonic flow associated to the current $T.$
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。
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。
The object of this survey is to give an overview on the topology of singularities of holomorphic foliation germs on $(\mathbb C^2,0)$.
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。
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 …
In this article we investigate the natural domain of definition of a holonomy map associated to a singular holomorphic foliation of the complex projective plane. We prove that germs of holonomy between algebraic curves can have large sets of singularities for the analytic continuation. In the Riccati context we provide examples with natural boundary and maximal sets of singularities. In the generic case we provide examples having at least a Cantor set of singularities and even a nonempty open set of singularities. The examples provided are based on the presence of sufficiently rich contracting…
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 $\…
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 this article, for holomorphic foliations of codimension one at $(\mathbb{C}^{3},0)$, we define the family of second type foliations. This is formed by foliations having, in the reduction process by blow-up maps, only well oriented singularities, meaning that the reduction divisor does not contain weak separatrices of saddle-node singularities. We prove that the reduction of singularities of a non-dicritical foliation of second type coincides with the desingularization of its set of separatrices.
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。
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.
The set $\mathbf{F}(3)$ of foliations of degree three on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $23$ on which acts $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$. The subset $\mathbf{FP}(3)$ of $\mathbf{F}(3)$ consisting of foliations of $\mathbf{F}(3)$ with a flat Legendre transform (dual web) is a Zariski closed subset of $\mathbf{F}(3)$. We classify up to automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$ the elements of $\mathbf{FP}(3)$. More precisely, we show that up to automorphism there are $16$ foliations of degree three wit…
We describe the topological types of leaves of generic logarithmic foliations on the complex projective plane. We prove that all leaves, except for a finite many are biholomorphic to $\mathbb{C}$ or homeomorphic to the surface known as Loch Ness monster.
Rigidity of singular de-Sitter tori with respect to their lightlike bi-foliation
本文研究了具有锥形奇点的洛伦兹面的刚性，证明了德西特环面由其光状双分层的拓扑等价类确定。
Let $\mathcal F$ be a holomorphic one-dimensional foliation on $\mathbb{P}^n$ such that the components of its singular locus $Σ$ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms of invariants of $\mathcal F$ and $C_i$, assuming that $\mathcal F$ is special along the $C_i$. Allowing just one nonzero dimensional component on $Σ$, we also prove results on when the foliation happens to be determined by its singular locus.
In this article, we prove two results. First, we construct a dense subset in the space of polynomial foliations of degree $n$ such that each foliation from this subset has a leaf with at least $\frac{(n+1)(n+2)}2-4$ handles. Next, we prove that for a generic foliation invariant under the map $(x, y)\mapsto (x, -y)$ all leaves have infinitely many handles.
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.
Motivated by the study of Hörmander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized functions on the underlying manifold, and on the leaves and their holonomy covers. This generalizes Schwartz kernel operators to singular foliations. We also define the algebra of smoothing operators in this context and prove that it is a two-sided ideal.
In this article we study polynomial logarithmic $q$-forms on a projective space and characterize those that define singular foliations of codimension $q$. Our main result is the algebraic proof of their infinitesimal stability when $q=2$ with some extra degree assumptions. We determine new irreducible components of the moduli space of codimension two singular projective foliations of any degree, and we show that they are generically reduced in their natural scheme structure. Our method is based on an explicit description of the Zariski tangent space of the corresponding moduli space at a given…
We study codimension one (transversally oriented) foliations $\fa$ on oriented closed manifolds $M$ having non-empty compact singular set $\sing(\fa)$ which is locally defined by Bott-Morse functions. We prove that if the transverse type of $\fa$ at each singular point is a center and $\fa$ has a compact leaf with finite fundamental group or a component of $\sing(\fa)$ has codimension $\ge 3$ and finite fundamental group, then all leaves of $\fa$ are compact and diffeomorphic, $\sing(\fa)$ consists of two connected components, and there is a Bott-Morse function $f:M \to [0,1]$ such that $f\col…
Let $\mathcal{F}$ be written as $ f^{*}\mathcal{G}$, where $\mathcal{G}$ is a foliation in $ {\mathbb P^2}$ with three invariant lines in general position, say $(XYZ)=0$, and $f:{\mathbb P^n}--->{\mathbb P^2}$, $f=(F^α_{0}:F^β_{1}:F^γ_{2})$ is a nonlinear rational map. Using local stability results of singular holomorphic foliations, we prove that: if $n\geq 3$, the foliation $\mathcal{F}$ is globally stable under holomorphic deformations. As a consequence we obtain new irreducible componentes for the space of codimension one foliations on $\mathbb P^n$. We present also a result which characte…
A codimension one singular holomorphic foliation is Newton non-degenerate if it satisfies the classical conditions of Kouchnirenko and Oka, in terms of its Newton polyhedra system. In this paper we prove that a foliation is Newton non-degenerate if and only if it admits a logarithmic reduction of singularities of a combinatorial nature.
Let $\mathcal{F}$ be written as $ f^{*}(\mathcal{G})$, where $\mathcal{G}$ is a $1$-dimensional foliation on $ {\mathbb P^{n-1}}$ and $f:{\mathbb P^n}--->{\mathbb P^{n-1}}$ a non-linear generic rational map. We use local stability results of singular holomorphic foliations, to prove that: if $n\geq 4$, a foliation $\mathcal{F}$ by complex surfaces on $\mathbb P^n$ is globally stable under holomorphic deformations. As a consequence, we obtain irreducible components for the space of two-dimensional foliations in $\mathbb P^n$. We present also a result which characterizes holomorphic foliations o…
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~$\mathbb{P}^2_{\mathbb C}.$
This paper contributes to the solution of the Poincare problem, which is to bound the degree of a (generalized algebraic) leaf of a (singular algebraic) foliation of the complex projective plane. The first theorem gives a new sort of bound, which involves the Castelnuovo--Mumford regularity of the singular locus of the leaf. The second theorem gives a bound in terms of two singularity numbers of the leaf: the total Tjurina number, and the number of non-quasi-homogeneous singularities. If such singularities are present, then this bound improves one due to du Plessis and Wall, at least when the …
We provide several applications of the minimal model program to the local and global study of co-rank one foliations on threefolds. Locally, we prove a singular variant of Malgrange's theorem, a classification of terminal foliation singularities and the existence of separatrices for log canonical singularities. Globally, we prove termination of flips, a connectedness theorem on lc centres, a non-vanshing theorem and some hyperbolicity properties of foliations.
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 \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…
In this work we consider holomorphic foliations of degree two on the projective plane $\mathbb{P}^2$ having an invariant line. In a suitable choice of affine coordinates these foliations are induced by a quadratic vector field over the affine part in such a way that the invariant line corresponds to the line at infinity. We say that two such foliations are topologically equivalent provided there exists a homeomorphism of $\mathbb{P}^2$ which brings the leaves of one foliation onto the leaves of the other and preserves orientation both on the ambient space and on the leaves. The main result of …
From pre-lamination to foliated plane
本文研究了平面奇异叶面与圆圈预层状结构之间的对应关系，确定了哪些预层状结构是叶面的无穷远边界，并构建了对应的唯一叶面。
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…
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 the singular set of a codimension one holomorphic foliations on $\mathbb{P}^3$. We find a local normal form of a codimension two component of the singular set that is not of Kupka type. We also determined the number of non-Kupka points immersed in a codimension two component of the singular set of a codimension one foliation on $\mathbb{P}^3$.
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 …
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。
We consider holomorphic foliations of dimension $k>1$ and codimension $\geq 1$ in the projective space $\mathbb{P}^n$, with a compact connected component of the Kupka set. We prove that, if the transversal type is linear with positive integers eigenvalues, then the foliation consist on the fibers of a rational fibration. As a corollary, if $\mathcal{F}$ is a foliation such that $dim(\mathcal{F})\geq cod(\mathcal{F})+2$ and has transversal type diagonal with different eigenvalues, then the Kupka component $K$ is a complete intersection and we get the same conclusion. The same conclusion holds i…
A holomorphic foliation on $\mathbb P^2_{\mathbb C}$, or a real analytic foliation on $\mathbb{P}^{2}_{\mathbb{R}},$ is said to be convex if its leaves other than straight lines have no inflection points. The classification of the convex foliations of degree $2$ on $\mathbb P^2_{\mathbb C}$ has been established in $2015$ by C.~\textsc{Favre} and J.~\textsc{Pereira}. The main argument of this classification was a result obtained in~$2004$ by~D.~\textsc{Schlomiuk} and N.~\textsc{Vulpe} concerning the real polynomial vector fields of degree $2$ whose associated foliation on $\mathbb{P}^{2}_{\math…
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