arXiv subjects

Adolfo Guillot

Publications and source records attributed to Adolfo Guillot.

13 recordsLinked to original sources

arXiv2026

Explicit integration, birational models and symmetries of Lins Neto's exceptional families of foliations

In 2002, Lins Neto introduced three remarkable one-parameter families of holomorphic foliations by curves on the complex projective plane, of degrees two, three, and four, whose properties established that the general form of the Poincar\'e Problem had no solution. These foliations are known to be birationally equivalent to certain quotients of linear foliations on abelian surfaces. We give explicit formulas for these birational equivalences, obtaining, in particular, parametrizations of the leaves of the foliations. For the families of degrees three and four, we determine explicit generators for the groups of birational transformations of the projective plane preserving them. For foliations in these families admitting a rational first integral, that is, for those whose parameter is an Eisenstein rational, we give a complete description of the nature and position of the singular points of a generic integral curve, and we present an algorithm that computes the rational first integral explicitly. Lins Neto's foliations can also be defined over algebraically closed fields of positive characteristic, and, in this setting, we characterize those that are algebraically integrable. Finally, we study the integrability of the reductions to fields of positive characteristic of some of the non-integrable foliations in the complex family.

math.AG↗

arXiv2025

Birational Geometry of Special Quotient Foliations and Chazy's Equations

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 Jonqui\`eres 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.

math.AG↗

arXiv2025

Uniformizable foliated projective structures along singular foliations

We consider holomorphic foliations by curves on compact complex manifolds, for which we investigate the existence of projective structures along the leaves varying holomorphically (foliated projective structures), 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 K\"ahler 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 any dimension, a foliation by curves of degree at least two, with only non-degenerate singularities, does not admit a strongly uniformizable foliated projective structure.

math.CV↗

arXiv2024

On Bureau's classification of quadratic differential equations in two variables free of movable critical points

As part of the efforts aimed at extending Painlev\'e and Gambier's work on second-order equations in one variable to first-order ones in two, in 1981, Bureau classified the systems of ordinary quadratic differential equations in two variables which are free of movable critical points (which have the Painlev\'e Property). We revisit this classification, which we complete by adding some cases overlooked by Bureau, and by correcting some of his arguments. We also simplify the canonical forms of some systems, bring the natural symmetries of others into their study, and investigate the birational equivalence among some of the systems in the class. Lastly, we study the birational geometry of Okamoto's space of initial conditions for Bureau's system VIII, in order to establish the sufficiency of some necessary conditions for the absence of movable critical points.

math.CA↗

arXiv2021

Foliated affine and projective structures

We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces.

math.DG↗

arXiv2019

Meromorphic vector fields with single-valued solutions on complex surfaces

We study ordinary differential equations in the complex domain given by meromorphic vector fields on Kähler compact complex surfaces. We prove that if such an equation has a maximal single valued solution with Zariski-dense image (in particular, if it has an entire one) then, up to a bimeromorphic transformation, either the vector field is holomorphic or it preserves a fibration.

math.CV↗

arXiv2019

On the multipliers at fixed points of self-maps of the projective plane

This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of $\mathbb{P}^2$ and quadratic homogeneous vector fields on $\mathbb{C}^3$, the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on $\mathbb{C}^3$ having exclusively single-valued solutions.

math.AG↗

arXiv2018

Quadratic Differential Equations in Three Variables without Multivalued Solutions: Part I

For ordinary differential equations in the complex domain, a central problem is to understand, in a given equation or class of equations, those whose solutions do not present multivaluedness. We consider autonomous, first-order, quadratic homogeneous equations in three variables, and begin the classification of those which do not have multivalued solutions.

math.CA↗

arXiv2014

A classification of locally homogeneous affine connections on compact surfaces

We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi-Civita connection of a Riemannian metric of constant curvature or the quotient of a translation-invariant connection in the plane.

math.DG↗

arXiv2013

Quasihomogeneous analytic affine connections on surfaces

We classify torsion-free real-analytic affine connections on compact oriented real-analytic surfaces which are locally homogeneous on a nontrivial open set, without being locally homogeneous on all of the surface. In particular, we prove that such connections exist. This classification relies in a local result that classifies germs of torsion-free real-analytic affine connections on a neighborhood of the origin in the plane which are quasihomogeneous, in the sense that they are locally homogeneous on an open set containing the origin in its closure, but not locally homogeneous in the neighborhood of the origin.

math.DG↗

arXiv2013

Vector fields, separatrices and Kato surfaces

We prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is compact). We also prove that, in a singular Stein surface endowed with a complete holomorphic vector field, a singular point of the surface where the zeroes of the vector field do not accumulate is either a quasihomogeneous or a cyclic quotient singularity. The proofs rely in a combinatorial description of the vector field on a resolution of the singular point based on previous work of Rebelo and the author. With the same tools, we reprove some facts about the classification of compact complex surfaces admitting holomorphic vector fields.

math.DS↗

arXiv2012

The geometry of Chazy's homogeneous third-order differential equations

Chazy studied a family of homogeneous third-order autonomous differential equations. They are those, within a certain class, admitting exclusively single-valued solutions. Each one of these equations yields a polynomial vector field in complex three-dimensional space. For almost all of these these vector fields, the Zariski closure of a generic orbit yields an affine surface endowed with a holomorphic vector field that has exclusively single-valued solutions. We classify these surfaces and relate this classification to recent results of Rebelo and the author.

math.DS↗

arXiv2010

Singular sets of holonomy maps for algebraic foliations

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 dynamics in the holonomy pseudogroup of the foliation.

math.DS↗