Spherical extension property no longer true for domains in algebraic variety with isolated singularity
type: Item
instance of: scholarly article
title: Spherical extension property no longer true for domains in algebraic variety with isolated singularity
zbMATH Open document ID: 1194.32022
author: Shanyu Ji
author: Cheng Zhan
author: Stephen Yau
published in: Science China. Mathematics
publication date: 2010-04-13T00:00:00Z
review text: Let \(D_1\) and \(D_2\) be simply connected bounded domains in \({\mathbb C}^n\) with real analytic spherical boundary. The boundary is called spherical if it is locally CR-equivalent to a portion of the unit sphere. By the generalization of the Riemann mapping theorem proved by \textit{S.-S. Chern} and \textit{S. Ji} [Ann. Math. 144, 421--439 (1996; Zbl 0872.32016)], every locally biholomorphic map between \(\partial D_1\) and \(\partial D_2\) extends to a biholomorphic map from \(D_1\) onto \(D_2\). If the boundary of the domains is algebraic, then the condition of simply connectedness can be removed [see \textit{S. M. Webster}, Invent. Math. 43, 53--68 (1977; Zbl 0348.32005)]. In this paper the authors show that for simply connected domains with algebraic spherical boundary in an algebraic variety with isolated singularities the above result does not hold. As a counterexample they exhibit a pair of domains in the algebraic variety \(\{z_1z_2=z_3^2\}\subset {\mathbb C}^3\).
reviewed by: Laura Geatti
Mathematics Subject Classification ID: 32V25
Mathematics Subject Classification ID: 32V15
Mathematics Subject Classification ID: 14R20
zbMATH DE Number: 5692425
MaRDI profile type: Publication
full work available at URL: s11425-010-0001-2
OpenAlex ID: W1976699438
cites work: On the Riemann mapping theorem
cites work: On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions
cites work: Global holomorphic extension of a local map and a Riemann mapping theorem for algebraic domains
cites work: The topology of normal singularities of an algebraic surface and a criterion for simplicity
DOI: 10.1007/S11425-010-0001-2
description: scientific article; zbMATH DE number 5692425