Local polar invariants and the Poincaré problem in the dicritical case
type: Item
instance of: scholarly article
title: Local polar invariants and the Poincaré problem in the dicritical case
zbMATH Open document ID: 1414.32030
author: Yohann Genzmer
author: Rogério S. Mol
published in: Journal of the Mathematical Society of Japan
publication date: 2019-01-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1608.07217
full work available at URL: https://projecteuclid.org/euclid.jmsj/1538380983
review text: The carefully written paper under review is of great interest to anyone studying singularities of plane holomorphic foliations. The review that follows cannot be made more precise without providing the numerous required details that the authors include. \par The authors make a further attempt at a complete explanation of the relation between the degree of a holomorphic foliation on the complex projective plane and the degree/genus of an invariant algebraic branch (i.e., the Poincaré Problem). Notwithstanding the existence of invariant curves of arbitrary degree for (necessarily dicritical) foliations of degree $1$ (e.g., $pxdy-qydx$, in a local chart), it might be possible to find conditions on the foliation providing a finer description of the relation between, say, the structure of the desingularization of the foliation and the multiplicity of the invariant curves at the singularities of the former. \par The authors pursue the path of \textit{local polar invariants}, which has proved fruitful in past research [\textit{N. Corral}, Ann. Inst. Fourier 53, No. 3, 787--814 (2003; Zbl 1032.32019); \textit{F. Cano}, \textit{N. Corral} and \textit{R. Mol} , ``Local polar invariants for plane singular foliations'', Expositiones Mathematicae (to appear)] on plane singular foliations, and prove a collection of relevant results relating those polar invariants (i.e., invariants associated to the generic \textit{polar curve} of a foliation) to other properties of the foliation. By using what they call a ``balanced equation'' of the separatrices of a germ of foliation $\mathcal{F}$, they show (for instance) that $\mathcal{F}$ is a generalized curve if and only if the ``polar excess'' (an arithmetic invariant defined in terms of the polar curve and the separatrices of $\mathcal{F}$) is $0$. They also relate the Gómez-Mont-Seade-Verjovsky index of $\mathcal{F}$ (see [\textit{X. Gómez-Mont} et al., Math. Ann. 291, No. 4, 737--752 (1992; Zbl 0725.32012)]) to that polar excess. Using…
reviewed by: Pedro Fortuny Ayuso
Mathematics Subject Classification ID: 32S65
zbMATH DE Number: 7009707
zbMATH Keywords: holomorphic foliations
zbMATH Keywords: singularities
zbMATH Keywords: invariant curves
zbMATH Keywords: Poincaré problem
zbMATH Keywords: GSV-index
MaRDI profile type: Publication
OpenAlex ID: W2895451928
arXiv ID: 1608.07217
cites work: On the classification of nilpotent singularities
cites work: Feuilletages holomorphes sur les surfaces complexes compactes
cites work: Some remarks on indices of holomorphic vector fields
cites work: Topological invariants and equidesingularization for holomorphic vector fields
cites work: Invariant varieties through singularities of holomorphic vector fields
cites work: Proximity inequalities and bounds for the degree of invariant curves by foliations of ℙ_{ℂ}²
cites work: Q4411321
cites work: The Poincaré problem in the nondicritical case
cites work: Localization of Baum-Bott residues, generalized curves and \(K\)-theory. I. Foliations in \(\mathbb{C}^2\)
cites work: On the Poincaré inequality for one-dimensional foliations
cites work: Holomorphic foliations in \(CP(2)\) having an invariant algebraic curve
cites work: Groupes d'automorphismes de $({\bbfC},0)$ et équations différentielles $ydy+\cdots =0$
cites work: Isolated invariant curves of a foliation
cites work: Bounds on leaves of one-dimensional foliations
cites work: The Poincaré problem, algebraic integrability and dicritical divisors
cites work: Rigidity for dicritical germ of foliation in ℂ2
cites work: The index of a holomorphic flow with an isolated singularity
cites work: Some examples for the Poincaré and Painlevé problems
cites work: Holonomie et intégrales premières
cites work: Q4407969
cites work: Classification analytique d'�quations diff�rentielles ydy +...=0 et espace de modules
cites work: Nilpotent singularities and first integrals
cites work: Meromorphic first integrals: Some extension results
cites work: Degree of the first integral of a pencil in \(\mathbb{P}^2\) defined by Lins Neto
cites work: Reduction of Singularities of the Differential Equation Ady = Bdx
DOI: 10.2969/JMSJ/76227622
description: scientific article