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In this paper we study differentiable equivalences of germs of singular holomorphic foliations in dimension two. We prove that the Camacho–Sad indices are invariant by such equivalences. We also prove that the Baum–Bott index is a differentiable invariant for some classes of foliations. As a corollary we show that generic degree two holomorphic foliations of P 2 are differentiably rigid. [ABSTRACT FROM AUTHOR]

We study germs of holomorphic distributions with "separated variables". In codimension one, a well know example of this kind of distribution is given by d z = (y 1 d x 1 - x 1 d y 1) + ⋯ + (y m d x m - x m d y m) , which defines the canonical contact structure on C P 2 m + 1 . Another example is the Darboux distribution d z = x 1 d y 1 + ⋯ + x m d y m , which gives the normal local form of any contact structure. Given a germ D of holomorphic distribution with separated variables in (C n , 0) , we show that there exists , for some κ ∈ Z ≥ 0 related to the Taylor coefficients of D , a holomorphic submersion H D : (C n , 0) → (C κ , 0) such that D is completely non-integrable on each level of H D . Furthermore, we show that there exists a holomorphic vector field Z tangent to D , such that each level of H D contains a leaf of Z that is somewhere dense in the level. In particular, the field of meromorphic first integrals of Z and that of D are the same. Between several other results, we show that the canonical contact structure on C P 2 m + 1 supports a Legendrian holomorphic foliation whose generic leaves are dense in C P 2 m + 1 . So we obtain examples of injectively immersed Legendrian holomorphic open manifolds that are everywhere dense. [ABSTRACT FROM AUTHOR]

An example is given of a curve divisor which is not the trace divisor of a foliation. [ABSTRACT FROM AUTHOR]

This paper deals with foliations by curves [s] of degree r ≥ 2 on P 2 with isolated singularities S, called non-degenerate if S is reduced and otherwise degenerate. Say that [s] is uniquely determined by a zero-dimensional Y ⊂ S if [s] is the unique foliation that vanishes on Y and say that Y is minimal for [s] if, moreover, the degree of Y is the minimal possible to do so. Previous work of the authors show that every non-degenerate foliation in degrees 2 ≤ r ≤ 5 does have a minimal subscheme and that the set of non-degenerate foliations of degree r ≥ 6 that have a minimal subscheme contains a Zariski-open subset of the space of foliations of degree r. For non-degenerate foliations [s] we present both characterizations and sufficient conditions for [s] to have a minimal subscheme. We also give examples of degenerate foliations of degree r ≥ 7 that do not have a minimal subscheme at all. [ABSTRACT FROM AUTHOR]

The space of holomorphic foliations of codimension one and degree d ≥ 2 in P n ( n ≥ 3 ) has an irreducible component whose general element can be written as a pullback F ∗ F , where F is a general foliation of degree d in P 2 and F : P n ⤏ P 2 is a general rational linear map. We give a polynomial formula for the degrees of such components. [ABSTRACT FROM AUTHOR]

For every integer k ≥ 3 we describe a new family of foliations of degree k with one singularity. We show that a very generic member of this family has trivial isotropy group and a line as its unique Darboux polynomial. [ABSTRACT FROM AUTHOR]

We fix a mistake in the argument leading to the proof that the family of foliations introduced in the paper does not have an algebraic solution apart from the line at infinity [ABSTRACT FROM AUTHOR]

Following Suwa, we study unfoldings of algebraic foliations and their relationship with families of foliations, making focus on those unfoldings related to trivial families. The results obtained in the study of unfoldings are then applied to obtain information on the structure of foliations on projective spaces. [ABSTRACT FROM AUTHOR]

We present new irreducible components of the space of codimension one holomorphic foliations on $$\mathbb P^{n}$$ , $$n\ge 3$$ . They are associated to pull-back by branched rational maps of foliations on $$\mathbb P^2$$ that preserve invariant lines. [ABSTRACT FROM AUTHOR]

It is proved, that a foliation on a modular curve given by the vertical trajectories of holomorphic differential corresponding to the Hecke eigenform is either the Strebel foliation or the pseudo-Anosov foliation. [ABSTRACT FROM AUTHOR]

In this article we prove in a new way that a generic polynomial vector field in $$\mathbb {C}^{2}$$ possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain. [ABSTRACT FROM AUTHOR]

We study neighborhoods of rational curves in surfaces with self-intersection number 1 that can be linearised. [ABSTRACT FROM AUTHOR]

We study compact invariant sets for holomorphic foliations on Stein manifold. As application, we show some dynamical properties concerning minimal sets (with singularities) of foliations and real analytic Levi-flat hypersurfaces in projective spaces. [ABSTRACT FROM AUTHOR]

We consider the set K( n, c, xdy − ydx) of codimension one holomorphic foliations on P, n ≥ 3, with normal Chern class c, and with a compact, connected Kupka set of radial transversal type. We will prove that foliations in this set, have a rational first integral and define an irreducible component of the space of foliations. [ABSTRACT FROM AUTHOR]

Let F be a holomorphic foliation on P by curves such that the components of its singular locus are curves C and points p . We compute the Baum-Bott indices BB ( F, C ) in terms of the main invariants of F and C . We also determine the sum of the BB ( F, p ) in terms of the same invariants.When φ corresponds to the determinant, the latter result generalizes, from special to all holomorphic foliations, a formula for the number of isolated singularities of F, counted with multiplicities. [ABSTRACT FROM AUTHOR]

Themain goal of ourwork is to showthat there exists a class of 2×2 Riemann problems for which the solution comprises a singlewave group for an open set of initial conditions. This wave group comprises a 1-rarefaction joined to a 2-rarefaction, not by an intermediate state, but by a doubly characteristic shock, 1-left and 2-right characteristic. In order to ensure that perturbations of initial conditions do not destroy the adjacency of the waves, local transversality between a composite curve foliation and a rarefaction curve foliation is necessary. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how some results about codimension one foliations in dimension three can be generalized to dimension two foliations in dimension n ≥ 4. [ABSTRACT FROM AUTHOR]

Certain families of manifolds which support Anosov flows do not support foliations which both are expanding under a dynamical system and have quasi-isometrically embedded leaves. [ABSTRACT FROM AUTHOR]

In this note we extend to arbitrary dimensions a couple of results due respectively to Mattei-Moussu and to Camara-Scardua in dimension 2. We also provide examples of singular foliations having a Siegel-type singularity and answering a question independently raised by Abate and by Genzmer. [ABSTRACT FROM AUTHOR]

We consider foliations on a germ of an isolated surface singularity. We give criteria for the existence of smooth separatrices in terms of the maximal cycle of a desingularization of the surface. We apply these criteria for some examples of quasi-homogeneous singularities, in particular rational double points; moreover, for these singularities we show for which cases there exist a smooth separatrix for foliations defined by sufficiently general derivations. [ABSTRACT FROM AUTHOR]

We define the polar curves and the polar family associated to a projective web and obtain some results about the geometry of the generic element of this family. We also deal with the particular case of foliations and prove the constancy of the topological embedded type of the generic polar. [ABSTRACT FROM AUTHOR]

We propound several problems concerning codimension one holomorphic foliations either in a local context (theory of singularities) or in a global situation (algebraic foliations). Each problem is illustrated by suitable examples. [ABSTRACT FROM AUTHOR]

In this paper we aim at the description of foliations having tangent sheaf T F with c ( T F) = c( T F) = 0 on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as a product, and that the Zariski closure of a general leaf of F is an Abelian variety. It turns out that the analytic type of the Zariski closures of leaves may vary from leaf to leaf. We discuss how this variation is related to arithmetic properties of the tangent sheaf of the foliation. [ABSTRACT FROM AUTHOR]

We consider Riccati foliations ℱ with hyperbolic leaves, over a finite hyperbolic Riemann Surface S, constructed by suspending a representation ρ: π( S) → PSL(2,ℂ) in a quasi-Fuchsian group. The foliated geodesic flow has a repeller-attractor dynamic with generic statistics µ and µ for positive and negative times, respectively. These measures have a common projection to a harmonic measure μ for the Riccati foliation. We describe μ, μ and μ in terms of the Patterson-Sullivan construction, and we show that the measures μ provide examples of the conformal harmonic measures introduced by M. Brunella. [ABSTRACT FROM AUTHOR]

This paper deals with codimension one (may be singular) foliations on compact Kälher manifoldswhose conormal bundle is assumed to be pseudo-effective. Using currents with minimal singularities, we show that one can endow the space of leaves with a metric of constant non positive curvature wich may degenerate on a 'rigidly' embedded invariant hypersurface. [ABSTRACT FROM AUTHOR]

In this paper we construct non-singular, locally-closed, algebraic varieties which are sets of foliations on ℂℙ of degree 2 with a certain degenerate singularity. We obtain the dimension and closure of these varieties. To do that we construct a stratification (based on GIT, see [7]) of the space of foliations with respect to the action by change of coordinates. We prove that the set of unstable foliations has two irreducible components. We have the following corollary: a foliation of degree 2 defined by a pencil of conics is unstable if and only if the pencil is unstable. Finallywe give another proof of the fact that there are only 4 foliations of degree 2 with a unique singular point (see [5]). [ABSTRACT FROM AUTHOR]

We exhibit a class of meromorphic functions in two variables conjugated to algebraic functions using the geometry of the foliation by level curves. [ABSTRACT FROM AUTHOR]

Let F: ℝ → ℝ be a polynomial local diffeomorphism and let S denote the set of not proper points of F. The Jelonek's real Jacobian Conjecture states that if codim( S) ≥ 2, then F is bijective. We prove a weak version of such conjecture establishing the sufficiency of a necessary condition for the bijectivity of F. [ABSTRACT FROM AUTHOR]

We consider singular foliations of codimension one on 3-manifolds, in the sense defined by André Haefliger as being Γ-structures. We prove that under the obvious linear embedding condition, they are Γ-homotopic to a regular foliation carried by an open book or a twisted open book. The latter concept is introduced for this aim. Our result holds true in every regularity C, r ≥ 1. In particular, in dimension 3, this gives a very simple proof of Thurston's 1976 regularization theorem without using Mather's homology equivalence. [ABSTRACT FROM AUTHOR]

We study the holonomy group of a holomorphic foliation in a surface along a compact leaf. It is shown that linearization of a neighborhood of the curve implies strong restrictions in those groups. [ABSTRACT FROM AUTHOR]

The Baum-Bott map associates to a foliation the Baum-Bott indexes of their singularities. In this paper we study the fibers of the Baum-Bott map in the space of foliations of degree two on the projective plane ℙ. In the main result we prove that its generic fiber contains exactly 240 orbits of the natural action of Aut(ℙ) onthespace of foliations. [ABSTRACT FROM AUTHOR]

In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a b-Poisson manifold as we will see in [9]. [ABSTRACT FROM AUTHOR]

This is an exposition of the subject, which was developed in the author's papers [19, 20]. Various results from the theory of foliations (cohomology, characteristic classes, deformations, etc.) are extended to subalgebroids of Lie algebroids that generalize the tangent integrable distributions. We also suggest a definition of foliated Courant algebroids and give some corresponding results and constructions. [ABSTRACT FROM AUTHOR]

We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy of vector fields and geometric entropy of foliations, and which can also be applied to singular objects, e.g. singular foliations, singular distributions, and Poisson structures. We show some basic properties for this entropy, including the additivity property, analogous to the additivity of Clausius-Boltzmann entropy in physics. In the case of Poisson structures, entropy is a new invariant of dynamical nature, which is related to the transverse structure of the characteristic foliation by symplectic leaves. [ABSTRACT FROM AUTHOR]

We study one-dimensional holomorphic foliations on products of complex projective spaces and present results giving the number of singularities, counting multiplicities, of a generic foliation, a criterion for a foliation to be Riccati and a Poincaré type inequality, relating degrees of foliations to degrees of hypersurfaces which are invariant by them. [ABSTRACT FROM AUTHOR]

In this paper we define closed partially conformal vector fields and use them to give a characterization of Riemannian manifolds which admit this kind of fields as some special warped products foliated by ( n − 1)-umbilical hypersurfaces. Examples are described in space forms. In particular, closed partially conformal vector fields in Euclidean spaces are associated to the most simple foliations given by hyperspheres, hyperplanes or coaxial cylinders. Finally, for manifolds admitting such vector fields, we impose conditions for a hypersurface to be ( n − 1)-umbilical, or, in particular, a leaf of the corresponding foliation. [ABSTRACT FROM AUTHOR]

Let F( z) = Re( P( z)) + h.o.t be such that M = ( F = 0) defines a germ of real analytic Levi-flat at 0 ∈ ℂ, n ≥ 2, where P ( z) is a homogeneous polynomial of degree k with an isolated singularity at 0 ∈ ℂ and Milnor number µ. We prove that there exists a holomorphic change of coordinate ϕ such that ϕ( M) = ( Re( h) = 0), where h( z) is a polynomial of degree µ + 1 and j ( h) = P. [ABSTRACT FROM AUTHOR]

We study foliations of space forms by complete hypersurfaces, under some mild conditions on its higher order mean curvatures. In particular, in Euclidean space we obtain a Bernstein-type theorem for graphs whose mean and scalar curvature do not change sign but may otherwise be nonconstant. We also establish the nonexistence of foliations of the standard sphere whose leaves are complete and have constant scalar curvature, thus extending a theorem of Barbosa, Kenmotsu and Oshikiri. For the more general case of r-minimal foliations of the Euclidean space, possibly with a singular set, we are able to invoke a theorem of Ferus to give conditions under which the non- singular leaves are foliated by hyperplanes. [ABSTRACT FROM AUTHOR]

The set ℱ(2; 2) of foliations of degree 2 on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension 14 on which acts Aut(ℙ(ℂ)). We classify, up to automorphisms of ℙ(ℂ), quadratic foliations with only one singularity. There are only four such foliations up to conjugacy; whereas three of them have a dynamic which can be easily described the dynamic of the fourth is still mysterious. This classification also allows us to describe the action of Aut(ℙ(ℂ)) on ℱ(2; 2). On the one hand we show that the dimension of the orbits is more than 6 and that there are exactly two orbits of dimension 6; on the other hand we obtain that the closure of the generic orbit in ℱ(2; 2) contains at least seven orbits of dimension 7 and exactly one orbit of dimension 6. [ABSTRACT FROM AUTHOR]

We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve in the 3-sphere or a Legendrian curve in the anti-de Sitter 3-space. We describe ruled Lagrangian surfaces and characterize the cyclic and ruled Lagrangian surfaces which are solutions to the self-similar equation of the Mean Curvature Flow. Finally, we give a partial result in the case of Hamiltonian stationary cyclic surfaces. [ABSTRACT FROM AUTHOR]

The polar curves of foliations $$ \mathcal{F} $$ having a curve C of separatrices generalize the classical polar curves associated to hamiltonian foliations of C. As in the classical theory, the equisingularity type ℘( $$ \mathcal{F} $$) of a generic polar curve depends on the analytical type of $$ \mathcal{F} $$, and hence of C. In this paper we find the equisingularity types ε( C) of C, that we call kind singularities, such that ℘( $$ \mathcal{F} $$) is completely determined by ε( C) for Zariski-general foliations $$ \mathcal{F} $$. Our proofs are mainly based on the adjunction properties of the polar curves. The foliation-like framework is necessary, otherwise we do not get the right concept of general foliation in Zariski sense and, as we show by examples, the hamiltonian case can be out of the set of general foliations. [ABSTRACT FROM AUTHOR]

We generalize Frobenius singular theorem due to Malgrange, for a large class of codimension one holomorphic foliations on singular analytic subsets of ℂ N. [ABSTRACT FROM AUTHOR]

We give a geometrical description of the principal curvature lines on an immersed canal hypersurface,and study the holonomy of the one-dimensional principal foliation that they define. [ABSTRACT FROM AUTHOR]

In this note, we recall the different notions of quasi-homogeneity for singular germs of holomorphic foliations in the plane presented in [6]. The classical notion of quasi-homogenity allude to those functions which belong to its own jacobian ideal. Given a foliation in the plane, asking that the equation of the separatrix set is a classical quasi-homogeneous function we obtain a natural generalization in the context of foliations. On the other hand, topological quasi-homogeneity is characterized by the fact that every topologically trivial deformation whose sepatrix family is analytically trivial is an analytically trivial deformation. We give an explicit example of a topological quasi-homogeneous foliation which is not quasi-homogeneous in the sense given above. [ABSTRACT FROM AUTHOR]

We classify the closed foliated 3-manifolds M, with codimeiision one foliations of nonexponential growth and which are homotopy equivalent to their classifying space BV. Then we construct arbitrarily "large" manifolds with fohations with of any growth type and satisfying π(M) = π1(BV). [ABSTRACT FROM AUTHOR]

We determine geometrically the number ∫Vmcm(TVm - i*Tf), associated to a smooth m-dimensional projective variety Vm, invariant by a one-dimensional holomorphic foliation F of Pcn, using polar divisors associated to the foliation and Bézout's theorem. [ABSTRACT FROM AUTHOR]

Consider oriented surfaces immersed in R[sup3]. Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature Κ, given by the product of the principal curvatures κ[sub1], κ[sub2] is positive. The leaves of the foliations are the lines of harmonic mean curvature, also called characteristic or diagonal lines, along which the normal curvature of the immersion is given by Κ/H, where H = (κ[sub1] + κ[sub2])/2 is the arithmetic mean curvature. That is, Κ/H= ((1/κ[sub1] + 1/κ[sub2]/2)[sup-1] is the harmonic mean of the principal curvatures κ[sub1], κ[sub2] of the immersion. The singularities of the foliations are the umbilic points and parabolic curves, where κ[sub1] = κ[sub2] and Κ = 0, respectively. Here are determined the structurally stable patterns of harmonic mean curvature lines near the umbilic points, parabolic curves and harmonic mean curvature cycles, the periodic leaves of the foliations. The genericity of these patterns is established. This provides the three essential local ingredients to establish sufficient conditions, likely to be also necessary, for Harmonic Mean Curvature Structural Stability of immersed surfaces. This study, outlined towards the end of the paper, is a natural analog and complement for that carried out previously by the authors for the Arithmetic Mean Curvature and the Asymptotic Structural Stability of immersed surfaces. [13, 14, 17], and also extended recently to the case of the Geometric Mean Curvature Configuration [15]. [ABSTRACT FROM AUTHOR]

Let φ a minimal affine Z[supp]-action on the torus T[supq], p ≥ 2 and q ≥ 1. The cohomology of φ (see definition below) depends on both the algebraic properties of the induced action on H[sup1](T[supq], Z) and the arithmetical properties of the translation cocycle. We give a Diophantine condition that characterizes those affine actions whose first cohomology group is finite dimensional. In this case it is necessarily isomorphic to R[supp]. Thus the R[supp]-action F[subφ] obtained by suspension of φ is parameter rigid, i.e., any other R[supp]-action with the same orbit foliation is smoothly conjugate to a reparametrization of F[subφ] by an automorphism of R[supp]. [ABSTRACT FROM AUTHOR]

Some properties of complex secondary classes are discussed. It is shown that the Godbillon-Vey class and the Bott class are related via complexification. [ABSTRACT FROM AUTHOR]

Let X be a variety over an algebraically closed field, η: Ω[sup 1, sub X] → L a one-dimensional singular foliation, and C ⊆ X a projective leaf of η. We prove that 2p[sub a](C) - 2 = deg(LC) + λ(C) - deg(C ∩ S) where p[sub a] (C) is the arithmetic genus, where λ(C) is the co-length in the dualizing sheaf of the subsheaf generated by the Kähler differentials, and where S is the singular locus of η. We bound λ(C) and deg(C ∩ S), and then improve and extend some recent results of Campillo, Carnicer, and de la Fuente, and of du Plessis and Wall. [ABSTRACT FROM AUTHOR]