Related papers: Convex foliations of degree 5 on the complex proje…

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.…

In this paper, we study homogeneous convex foliations on the complex projective plane $\mathbb{P}^2$. A foliation is called convex if all of its leaves, except straight lines, have no inflection points, and such foliations form a Zariski…

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…

Let $\mathcal{F}$ denote a singular holomorphic foliation on $\mathbb{P}^2$ having a finite automorphism group $\mbox{aut}(\mathcal{F})$. Fixed the degree of $\mathcal{F}$, we determine the maximal value that $|\mbox{aut}(\mathcal{F})|$ can…

We consider the existence of symplectic and conformal symplectic codimension-one foliations on closed manifolds of dimension at least 5. Our main theorem, based on a recent result by Bertelson-Meigniez, states that in dimension at least 7…

We prove a complete classification of degree-$2$ foliations on $\mathbb{P}^n$ in any dimension, assuming they are not algebraically integrable. If $\mathcal{F}$ is such a foliation, then either $\mathcal{F}$ is the linear pull-back of a…

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…

Let R be a commutative ring with 1. We prove that every homogeneous polynomial f(x_0,x_1,x_2) in R[x_0,x_1,x_2] up to degree 5 admits a linear Pfaffian R-representation. We believe that conceptually we give the shortest self-contained proof…

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…

The present paper is a continuation of [13], [14] of the authors. Specifically, the paper considers the MD5-foliations associated to connected and simply connected MD5-groups such that their Lie algebras have 4-dimensional commutative…

The aim of this work is to study the foliations on the complex projective plane with flat \textsc{Legendre} transform (dual web). We establish some effective criteria for the flatness of the dual $d$-web of a homogeneous foliation of degree…

The set $\mathbf{F}(d)$ of foliations of degree $d$ on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $(d+2)^2-2$ on which acts $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$. The…

In this paper, we study holomorphic foliations of degree four on complex projective space $\mathbb{P}^n$, where $n\geq 3$, with a special focus on obtaining a structural theorem for these foliations. Furthermore, for a foliation…

Let $\mathcal{F}_d(\mathbb{P}^n)$ be the space of all singular holomorphic foliations by curves on $\mathbb{P}^n$ ($n \geq 2$) with degree $d \geq 1.$ We show that there is subset $\mathcal{S}_d(\mathbb{P}^n)$ of…

Let C be a real nonsingular affine curve of genus one, embedded in affine n-space, whose set of real points is compact. For any polynomial f which is nonnegative on C(R), we prove that there exist polynomials f_i with f \equiv \sum_i f_i^2…

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…

Geometric conditions are given so that the leafwise reduced cohomology is of infinite dimension, specially for foliations with dense leaves on closed manifolds. The main new definition involved is the intersection number of subfoliations…

We study analytic deformations and unfoldings of holomorphic foliations in complex projective plane $\mathbb{C}P(2)$. Let $\{\mathcal{F}_t\}_{t \in \mathbb{D}_{\epsilon}}$ be topological trivial (in $\mathbb{C}^2$) analytic deformation of a…

Making use of the extended flux homomorphism on the group of symplectomorphisms of a closed oriented surface of genus at least 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently…

A holomorphic pre-foliation $\mathscr{F}=\mathcal{C}\boxtimes\mathcal{F}$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is the data of a reduced complex projective curve $\mathcal{C}$ of $\mathbb{P}^{2}_{\mathbb{C}}$ and a holomorphic foliation…