The works of Brunella and Santos have singled out three special singular holomorphic foliations on projective surfaces having invariant rational nodal curves of positive self-intersection. These foliations can be described as quotients of foliations on some rational surfaces under cyclic groups of transformations of orders three, four, and six, respectively. Through an unexpected connection with the reduced Chazy IV, V and VI equations, we give explicit models for these foliations as degree-two foliations on the projective plane (in particular, we recover Pereira's model of Brunella's foliation). We describe the full groups of birational automorphisms of these quotient foliations, and, through this, produce symmetries for the reduced Chazy IV and V equations. We give another model for Brunella's very special foliation, one with only non-degenerate singularities, for which its characterizing involution is a quartic de Jonquières one, and for which its order-three symmetries are linear. Lastly, our analysis of the action of monomial transformations on linear foliations poses naturally the question of determining planar models for their quotients under the action of the standard quadratic Cremona involution; we give explicit formulas for these as well.

Abstract Radial germs of holomorphic foliations in dimension two have a characteristic property: they are the only singular foliations whose reduction of singularities has no singular points. We also know that they are desingularized by a single dicritical blowing-up. Let us say that a foliated space $$(({\mathbb {C}}^3,{\textbf{0}}),E,{\mathcal {F}})$$ ( ( C 3 , 0 ) , E , F ) is almost radial when it has a reduction of singularities without singular points; it will be “radial” under a certain additional condition on the morphism of reduction of singularities. We show that the radial condition corresponds to the “open book” situation. We end the paper with a discussion on the general almost radial case.

Abstract In this paper, we present a series of seemingly unrelated results of Complex Analysis which are, in fact, connected via a different approach to their proofs using the results of Errett Bishop of volumes, extensions, and limits of analytic varieties. We start with a brief introduction to the tools developed by Bishop and show their usefulness by proving Chow’s theorem via a technique suggested a long time ago in a beautiful book by Gabriel Stolzenberg, then we show some of the relationships between the theory of analytic subsets and classical results of complex-analytic functions. We finish with the original contributions of the paper which consist of applications of these tools to the theory of holomorphic foliations with alternative and, we believe, simpler proofs to Edwards, Millet, and Sullivan’s impactful result for foliations with compact leaves in the case of complex foliations in Kähler manifolds and J. V. Pereira’s global stability result for holomorphic foliations on compact Kähler manifolds.

We prove that on the product of two elliptic curves a generic nonsingular turbulent holomorphic foliation does not admit any transversely holomorphic projective structure.

In this work, we study inequalities and enumerative formulas for flags of Pfaff Systems on [Formula: see text]. More specifically, we establish a bound for the number of independent twisted [Formula: see text]-forms that leave invariant a one-dimensional holomorphic foliation and deduce inequalities that relate the degrees in the flags, which can be interpreted as a version of the Poincaré problem for flags. Moreover, by restricting to a flag of specific holomorphic foliations/distributions, we obtain inequalities involving the degrees. As a consequence, we obtain stability results for the tangent sheaf of some rank two holomorphic foliations/ distributions.

Abstract We consider holomorphic foliations by curves on compact complex manifolds, for which we investigate the existence of foliated projective structures (projective structures along the leaves varying holomorphically) that satisfy particular uniformizability properties. Our results show that the singularities of the foliation impose severe restrictions for the existence of such structures. A foliated projective structure separates the singularities of a foliation into parabolic and non-parabolic ones. For a strongly uniformizable foliated projective structure on a compact Kähler manifold, the existence of a single non-degenerate, non-parabolic singularity implies that the foliation is completely integrable. We establish an index theorem that imposes strong cohomological restrictions on the foliations having only non-degenerate singularities that support foliated projective structures making all of them parabolic. As an application of our results, we prove that, on a projective space of arbitrary dimension, a foliation by curves of degree at least two, with only non-degenerate singularities, does not admit a strongly uniformizable foliated projective structure.

We are interested in characterizing the holonomy maps associated to integral curves of non-degenerate singularities of holomorphic vector fields. Such a description is well-known in dimension 2 where is a key ingredient in the study of reduced singularities. The most intricate case in the 2 dimensional setting corresponds to (Siegel) saddle singularities. This work treats the analogous problem for saddles in higher dimension. We show that any germ of holomorphic biholomorphism, in any dimension, can be obtained as the holonomy map associated to an integral curve of a saddle singularity. A natural question is whether we can prescribe the linear part of the saddle germ of vector field provided the holonomy map. The answer to this question is known to be positive in dimension 2. We see that this is not the case in higher dimension. In spite of this, we provide a positive result under a natural condition for the holonomy map.

Abstract Exponential actions defined by vector configurations provide a universal framework for several constructions of holomorphic dynamics, non‐Kähler complex geometry, toric geometry and topology. These include leaf spaces of holomorphic foliations, intersections of real and Hermitian quadrics, the quotient construction of simplicial toric varieties, LVM and LVMB manifolds, complex‐analytic structures on moment‐angle manifolds and their partial quotients, reviewed in this survey. In all cases, the geometry and topology of the appropriate quotient object can be described by combinatorial data including a pair of Gale dual vector configurations.

We investigate the structure of the [Formula: see text]-divisor associated with the Jouanolou foliation and we show, under certain conditions, that it can be irreducible or have a [Formula: see text]-factor. Furthermore, we study the reduction modulo [Formula: see text] of foliations and explore its applications to problems involving holomorphic foliations. As an application, we provide a new proof, via reduction modulo 2, of the fact that the Jouanolou foliation on the complex projective plane of odd degree, under specific arithmetic conditions, has no algebraic invariant curves.

Holomorphic Foliations of Degree Four on the Complex Projective Space

We prove an extension criterion for codimension one foliations on projective hypersurfaces based on the degree of the foliation and the degree of the hypersurface, and we ensure, in some instances, an isomorphism between the corresponding spaces of foliations. We also present some examples of foliations that do not satisfy the extension criterion and do not extend.

En este artículo, investigamos el problema de la existencia de foliaciones holomorfas en variedades de Hopf de dimensión 3, con un enfoque particular en las variedades de tipo excepcional. Las variedades de Hopf, al ser variedades complejas compactas y no kählerianas, ofrecen un entorno fértil para el análisis de fenómenos no triviales en el estudio de foliaciones holomorfas. En particular, estas variedades presentan estructuras geométricas que permiten la aparición de comportamientos dinámicos complejos, lo que las convierte en un caso de especial interés.

In this work, we study actions of the Lie group SL(2,C) on a complex manifold of dimension three or higher. It is demonstrated that these types of actions induce three complete holomorphic vector fields, one of which is periodic, and that there exists a particular relationship between them, given by the Lie bracket, which generates a singular holomorphic foliation of codimension two. Subsequently, the types of singularities are classified, and the normal forms of these vector fields are obtained in a neighborhood of each singular point of the foliation.

In this paper, we study the analytic classification of a class of nilpotent singularities of holomorphic foliations in (C2,0), those exhibiting a Poincaré-Dulac type singularity in their reduction process. This analytic classification is based in the holonomy of a certain component of the exceptional divisor. Finally, as a consequence, we show that these singularities exhibit a formal analytic rigidity.

Hyperbolic entropy for harmonic measures on singular holomorphic foliations

In this work we use our previous results on the topological classification of generic singular foliation germs on ( C 2 , 0 ) (\mathbb {C}^{2},0) to construct complete families: after fixing the semi-local topological invariants we prove the existence of a minimal family of foliation germs that contains all the topological classes and such that any equisingular global family with parameter space an arbitrary complex manifold factorizes through it.

Abstract We study the quotient variety of the space of foliations on ℂ ⁢ ℙ 2 {\mathbb{CP}^{2}} of degree 2 up to change of coordinates. We find the intersection Betti numbers of this variety. As a corollary, we have that these intersection Betti numbers coincide with the intersection Betti numbers of the quotient variety of quartic plane curves. Finally, we give an explicit isomorphism between the space of foliations of degree 2 with different singular points, without invariant lines and the space of smooth quartic plane curves.

Holomorphic foliation associated with a semi-positive class of numerical dimension one

These notes are a slightly enlarged version of my habilitation thesis, where our research interest and main results in the past few years are summarized. Most of the discussion revolves around complex ordinary differential equations and their underling foliations, singularity theory and dynamical systems. Compared to the original text, a section containing some background material on holomorphic foliations was added. Also some new results obtained in the past three years that are in line with the one presented in the habilitation were included.

The Suwa method for computing versal unfoldings of holomorphic singular foliations is considered from the point of view of computational complex analysis. Based on the theory of Grothendieck local duality on residues, an effective algorithm of computing a first order infinitesimal versal unfoldings of codimension one complex analytic singular foliations is obtained. As an application of our approach, we give an effective method for computing universal unfoldings of germs of meromorphic functions.