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 singularities on the infinity line. Also we prove that a finitely generated subgroup of $\mathrm{Diff}(\mathbb{C}^n,0)$ with polynomial growth is solvable.
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 …
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 $\ge 3$.
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.
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.
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.
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…
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…
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 …
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 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.
In this paper I give a condition, in terms of boundary behaviour, that forces the infinitesimal generator of a one-parameter semigroup of holomorphic maps to vanish.
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 …
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.
In this paper we address the issues of absolute continuity for the center foliation (as well as the disintegration on the non-absolute continuous case) and rigidity of volume preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov on $\mathbb T^3$. It is shown that the disintegration of volume on center leaves may be neither atomic nor Lebesgue. It is also obtained results concerning the atomic disintegration. Moreover, the absolute continuity of the center foliation does not imply smooth conjugacy with its linearization. Imposing stronger conditions besides absolute contin…
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.
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 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.
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.
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.
A foliation is R-covered if the leaf space in the universal cover is homeomorphic to the real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to such a foliation. If there are two, then the foliation is weakly conjugate to the the stable foliation of an R-covered Anosov flow. The proof uses the universal circle to R-covered foliations.
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.
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 $\leq n/k.$
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 consider a class of foliations on the complex projective plane that are determined by a quadratic vector field in a fixed affine neighborhood. Such foliations, as a rule, have an invariant line at infinity. Two foliations with singularities on $\mathbb C P^2$ are topologically equivalent provided that there exists a homeomorphism of the projective plane onto itself that preserves orientation both on the leaves and in $\mathbb C P^2$ and brings the leaves of the first foliation to that of the second one. We prove that a generic foliation of this class may be topologically equivalent to but a…
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 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 $X$.
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 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 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.
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.
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…
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.
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.
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.
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…
