On germs of holomorphic foliations admitting sectorial first integrals
type: Item
instance of: scholarly article
title: On germs of holomorphic foliations admitting sectorial first integrals
zbMATH Open document ID: 1475.37054
author: Yanyan Li
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2021-05-03T00:00:00Z
review text: Let \(\mathcal{F}\) be a holomorphic foliation in a neighborhood \(U\subset \mathbb{C}^{2}\) of the origin \(0\). Assume that singular set of \(\mathcal{F}\) is \(0\) and that \(\mathcal{F}\) has at most a finite amount of separatrices through it (\(0\) is a non-dicritical singularity). Recall that a separatrix \(\Gamma\) through \(0\) is a germ of analytic irreducible curve which contains \(0\) and such that \(\Gamma\setminus\{0\}\) is contained in a leaf of \(\Gamma\). According to reduction of singularities, after a finite composition of blowing ups \(\pi\) over \(0\), the pull-back foliation \(\pi^{\ast}\mathcal{F}\) only has a finite amount of singularities, which are over the exceptional divisor \(E=\pi^{-1}(0)\). If all these simple singularities have linear part with two eigenvalues different from zero (non-degenerate singularities), \(\mathcal {F}\) is said to be a non-dicritical generalized curve. Here the authors mainly study a non-dicritical generalized curve \(\mathcal{F}\) when for a given separatrix \(\Gamma\) through \(0\) and any transverse disk \(\Sigma_{p}\subset U\) to \(\mathcal{F}\) with \(\Sigma_{p}\cap \Gamma=\{p\}:\) (1) There exists a sector \(S\subset \Sigma_{p}\) with vertex \(p\) and a holomorphic function \(\varphi:S\to\mathbb{C}\) constant on the sets \(L\cap S\) where \(L\) is a leaf of \(\mathcal{F}\) (\(\varphi\) is a sectorial first integral); (2) There exists a formal power series \(\tilde{\varphi}={\sum}_{i=0}^{\infty}a_{j}z^{j} \) such that for any proper subsector \(S'\subset S\) and \(k>0\), there exists a constant \(A_{k}\) verifying that \(|\varphi(z)-{\sum}_{j=0}^{k-1}a_{j}z^{j}|\leq A_{k}{|z|}^{k}\) in \(S'\) (\(\varphi\) admits \(\tilde{\varphi}\) as asymptotic expansion). They prove in the main result (Theorem A) that such a \(\mathcal{F}\) always admits a holomorphic first integral in a neighbourhood of \(0\in\mathbb{C}^{2}\). To prove it, they first consider the simplest case where \(\mathcal{F}\) has a non-degenerate …
reviewed by: Alvaro Bustinduy
Mathematics Subject Classification ID: 37F75
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 32M25
Mathematics Subject Classification ID: 32S65
zbMATH DE Number: 7342840
zbMATH Keywords: holomorphic vector field
zbMATH Keywords: separatrix
zbMATH Keywords: holonomy techniques
Wikidata QID: Q114219072
MaRDI profile type: Publication
full work available at URL: s40590-021-00344-0
OpenAlex ID: W3138985646
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DOI: 10.1007/S40590-021-00344-0
description: scientific article; zbMATH DE number 7342840