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Maycol Falla Luza

Publications and source records attributed to Maycol Falla Luza.

17 recordsLinked to original sources
arXiv2026

The geography of Chern slopes with prescribed fundamental group

Let $G$ be the topological fundamental group of a nonsingular complex projective surface. Troncoso and Urz\'ua proved that the Chern slopes $c_1^2/c_2$ of minimal surfaces of general type $S$ with $\pi_1(S)\simeq G$ are dense in $[1,3]$, and left $[1/3,1)$ open. We prove that they are dense in $[1/2,3]$, an interval that cannot be enlarged without contradicting either a theorem of Mendes Lopes and Pardini or Reid's conjecture. We prove more: the slopes of such surfaces with $K_S$ ample are dense in $[1/2,2]$, the first case in which the conjecture of Troncoso and Urz\'ua on ample canonical classes is established. The tool is an exact ampleness criterion for their product construction, which shows in particular that their own surfaces never have ample canonical class, whatever the defining sections.

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arXiv2026

Neighborhoods of curves with a prescribed number of foliations

Given a connected projective curve $C \subset \mathbb{P}^n$, $n \geq 2$, and an integer $0 \leq \ell \leq n$, we construct an n-dimensional (non compact) complex manifold, obtained as a neighborhood of an embedded copy of $C$, which carries exactly $\ell$ codimension one holomorphic foliations; moreover, every codimension one distribution on it is one of these foliations. We also determine the field of meromorphic functions of these manifolds: it can be prescribed to be $\mathbb{C}$ or a purely transcendental extension of transcendence degree one, and no larger field is possible as soon as the number of foliations is finite. This extends to arbitrary dimension, and refines, previous constructions of neighborhoods of curves in surfaces without foliations or without non-constant meromorphic functions.

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arXiv2026

Homogeneous pre-foliations of co-degree one and degree four on the projective plane

We classify, up to projective automorphism, all homogeneous pre-foliations of co-degree $1$ and degree $4$ on the complex projective plane $\Ptwo$ whose Legendre transform defines a flat $4$-web. The classification is organized according to the type of the underlying homogeneous foliation $\Hcal$ of degree~$3$, distinguishing the cases $\deg(\Tcal_{\Hcal})=2$, $3$, and~$4$. The case $\deg(\Tcal_{\Hcal})=2$ was treated by Bedrouni, while the cases $\deg(\Tcal_{\Hcal})=3$ and $\deg(\Tcal_{\Hcal})=4$ are completed here. The proof combines Bedrouni's curvature-holomorphy criteria with explicit normal forms and symbolic computation; the result yields a finite list of explicit one-forms, parametrized by the ramification data of the Gauss map of~$\Hcal$, all displayed explicitly in the article (the few cases whose parameters are roots of higher-degree polynomials being written out in full in Appendix A).

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arXiv2026

Formulae for indices of holomorphic foliations via reduction of singularities

We study numerical invariants associated with the reduction of singularities of holomorphic foliation germs on $(\mathbb{C}^2, 0)$. Building on our previous work on generalized curve foliations, we extend explicit formulas for several fundamental invariants to arbitrary foliations. In particular, we provide general expressions for the discrepancy vector, the Milnor and intrinsic Milnor numbers, and classical indices along a separatrix as Camacho-Sad, Variation, G\'omez-Mont-Seade-Verjovsky and also the Baum-Bott index. These extensions require a careful analysis of the contributions of saddle-nodes arising in the desingularization process. As applications, we recover results of Brunella and Cavalier-Lehmann, as well as a related statement appearing in [8], within a unified and purely numerical framework. Furthermore, we obtain intrinsic characterizations of generalized curve foliations in terms of indices and of second type foliations in terms of the discrepancy vector.

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arXiv2025

Indices of holomorphic foliations and the bifurcation conjecture

In this paper, we revisit local invariants (G\'omez-Mont-Seade-Verjovsky, variation, Camacho-Sad and Baum-Bott indices) associated with singular holomorphic foliations on $(\mathbb{C}^2 , 0)$ and we provide semi-global formulas for them in terms of the reduction of singularities of the foliation. A key technical ingredient is the Cholesky-type factorization of the intersection matrix of the exceptional divisor, which allows for an explicit control of multiplicities and indices along the resolution process. Using this factorization, we express the Milnor number and other indices as quadratic forms in intersection vectors associated to balanced divisors introduced by Y. Genzmer. As a main application, we address a conjecture posed by A. Szawlowski concerning pencils of plane holomorphic germs. We prove that the excess of Milnor numbers along the pencil is precisely captured by the invariants derived from our formulas, thereby confirming the conjecture in full generality. This also yields a new expression for the dimension of the parameter space of universal unfoldings of meromorphic functions in the sense of T. Suwa.

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arXiv2025

Homogeneous Convex Foliations of degree 6

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 closed subset in the space of degree $d$ foliations on $\mathbb{P}^2$. Using projective duality, every foliation can be associated with a $d$-web on the dual plane via its Legendre transform, and it is known that the Legendre transform of a homogeneous convex foliation is flat. Our first main result provides a classification of homogeneous convex foliations admitting exactly three radial singularities on the line at infinity. As a second result, we complete the classification of convex homogeneous foliations of degree $6$, extending previous classifications in degrees $4$ and $5$.

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arXiv2024

Webs Generated by Products of convex and homogeneous Foliations on $\mathbb{P}^2$

This paper investigates flat webs on the projective plane. We present two methods for constructing such webs: the first involves taking the product of finitely many convex reduced foliations and invariant lines, while the second consists of taking the product of finitely many convex homogeneous foliations and invariant lines. In both cases, we demonstrate that the dual web is flat.

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arXiv2024

Non-algebraizable neighborhoods of curves

We provide several families of compact complex curves embedded in smooth complex surfaces such that no neighborhood of the curve can be embedded in an algebraic surface. Different constructions are proposed, by patching neighborhoods of curves in projective surfaces, and blowing down exceptional curves. These constructions generalize examples recently given by S. Lvovski. One of our non algebraic argument is based on an extension theorem of S. Ivashkovich.

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arXiv2024

Distributions and Legendrian foliations in dimension 3

We study the field of rational first integrals of distributions. We show that for a distribution on 3 dimensional manifolds there exists a tangent vector field with the same field of first integrals. We also show a similar result for integrable distributions in any dimension.

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arXiv2023

Submanifolds with ample normal bundle

We construct germs of complex manifolds of dimension $m$ along projective submanifolds of dimension $n$ with ample normal bundle and without non-constant meromorphic functions whenever $m \geq 2n$. We also show that our methods do not allow the construction of similar examples when $m < 2n$ by establishing an algebraicity criterion for foliations on projective spaces which generalizes a classical result by Van den Ven characterizing linear subspaces of projective spaces as the only submanifolds with split tangent sequence.

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arXiv2022

Distributions, first integrals and Legendrian foliations

We study germs of holomorphic distributions with "separated variables'. In codimension one, a well know example of this kind of distribution is given by the canonical contact structure on $\mathbb{P}^{2m+1}$ . Another example is the Darboux distribution, which gives the normal local form of any contact structure. Given a germ $D$ of holomorphic distribution with separated variables in $(\mathbb{C}^n,0)$, we show that there exists , for some $\kappa \in \mathbb{Z}_{\geq 0}$ related to the Taylor coefficients of $D$, a holomorphic submersion $H_{D}: (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^{\kappa},0)$ 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.

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arXiv2020

Neighborhoods of rational curves without functions

We prove the existence of (non compact) complex surfaces with a smooth rational curve embedded such that there does not exist any formal singular foliation along the curve. In particular, at arbitray small neighborhood of the curve, any meromorphic function is constant. This implies that the Picard group is not countably generated.

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arXiv2017

Projective structures, neighborhoods of rational curves and Painlev'e equations

We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification, existence of normal forms, pencil/fibration decomposition, infinitesimal symmetries. We deduce some transcendental result about Painlev'e equations.

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arXiv2015

Extactic divisors for webs and lines on projective surfaces

Given a web (multi-foliation) and a linear system on a projective surface we construct divisors cutting out the locus where some element of the linear system has abnormal contact with the leaf of the web. We apply these ideas to reobtain a classical result by Salmon on the number of lines on a projective surface. In a different vein, we investigate the number of lines and of disjoint lines contained in a projective surface and tangent to a contact distribution.

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arXiv2015

Foliations and webs inducing Galois coverings

We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on $\mathbb P^2$ admitting continuous symmetries, obtaining a complete classification of Galois homogeneous foliations.

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arXiv2011

Characteristic Numbers and invariant subvarieties for Projective Webs

We define the characteristic numbers of a holomorphic k-distribution of any dimension on $mathbb P^n$ and obtain relations between these numbers and the characteristic numbers of an invariant subvariety. As an application we bound the degree of a smooth invariant hypersurface.

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