Lipschitz regular complex algebraic sets are smooth
type: Item
instance of: scholarly article
title: Lipschitz regular complex algebraic sets are smooth
zbMATH Open document ID: 1338.14008
author: Lev Birbrair
author: A. Fernandes
author: Dũng Tráng Lê
author: J. Edson Sampaio
published in: Proceedings of the American Mathematical Society
publication date: 2016-03-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1404.5678
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32S50
zbMATH DE Number: 6549101
zbMATH Keywords: Lipschutz regular
zbMATH Keywords: smooth algebraic set
MaRDI profile type: Publication
OpenAlex ID: W1946711731
arXiv ID: 1404.5678
cites work: Tangent spaces and Gromov-Hausdorff limits of subanalytic spaces
cites work: On normal embedding of complex algebraic surfaces
cites work: Normal embeddings of semialgebraic sets
cites work: The topology of normal singularities of an algebraic surface and a criterion for simplicity
cites work: Cones in Complex Affine Space are Topologically Singular
cites work: Beyond Mumford's theorem on normal surfaces
DOI: 10.1090/PROC/12783
review text: A closed subset \(X\subset\mathbb{R}^n\) is called topologically regular (resp. Lipschitz regular) at \(x_0\in X\) if there exists a homeomorphism (resp. a subanalytic bi-Lipschitz homeomorphism) \(\Phi:B\to U\) where \(B\) is a Euclidean open ball and \(U\) is an open neighbourhood of \(x_0\) in \(X\). \(X\) is called topologically (resp. Lipschitz) regular if \(X\) is topologically (resp. Lipschitz) regular at any \(x\in X\). Mumford proved that a topologically regular complex algebraic surface in \(\mathbb{C}^3\) with an isolated singular point is smooth ([\textit{D. Mumford}, Publ. Math., Inst. Hautes Étud. Sci. 9, 5--22 (1961; Zbl 0108.16801)]).NEWLINENEWLINEIt is proved that any Lipschitz regular complex algebraic set is smooth. Note that the assumptions \(\dim(X)=2\) and isolated singularities are not needed.
reviewed by: Gerhard Pfister
description: scientific article; zbMATH DE number 6549101