Invariant holomorphic mappings
type: Item
instance of: scholarly article
title: Invariant holomorphic mappings
zbMATH Open document ID: 0901.32017
DOI: 10.1007/BF02921598
author: John P. D'Angelo
published in: The Journal of Geometric Analysis
publication date: 1998-12-02T00:00:00Z
review text: We investigate holomorphic mappings that are invariant under a finite subgroup of the unitary group on complex Euclidean space. Given such a group \(\Gamma\), and a real-analytic hypersurface \(M\) invariant under \(\Gamma\), we consider what hypersurfaces are possible targets for a non-constant \(\Gamma\)-invariant holomorphic mapping. We construct from a defining function \(r\) for \(M\) a \(\Gamma\)-invariant real-valued real-analytic function \(\Phi_{\Gamma,r}\), and use it to define an invariant holomorphic mapping taking \(M\) to a hyperquadric. We obtain general properties of \(\Phi_{\Gamma,r}\), and compute it for certain cyclic groups when \(M\) is the sphere \(S^3\). Its coefficients must then be integers; we determine them when \(S^3/ \Gamma\) is the lens space \(L(p,p-1)\), and find the target hyperquadrics. We also discuss a conjecture relating embedding dimensions of proper mappings between balls and the properties of invariant mappings.
Mathematics Subject Classification ID: 32H02
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 32M99
zbMATH DE Number: 1160132
zbMATH Keywords: \(\Gamma\)-invariant real-valued real-analytic function
zbMATH Keywords: hypersurfaces
zbMATH Keywords: invariant holomorphic mapping
zbMATH Keywords: cyclic groups
zbMATH Keywords: lens space
zbMATH Keywords: embedding dimensions
zbMATH Keywords: invariant mappings
MaRDI profile type: Publication
cites work: Q4172303
cites work: Polynomial proper maps between balls
cites work: Spherical space forms, CR-mappings, and proper maps between balls
cites work: Maps from the two-ball to the three-ball
cites work: Proper holomorphic maps from balls
cites work: Extending proper holomorphic mappings of positive codimension
cites work: Q4024771
cites work: Q5546515
full work available at URL: https://doi.org/10.1007/bf02921598
OpenAlex ID: W2094245976
Recommended article: Q4347747
Recommended article: Q4397659
Recommended article: Q3635360
Recommended article: Q4444979
Recommended article: Q4796249
Recommended article: Invariant distances and fixed points of holomorphic mappings
Recommended article: Invariant forms on complex manifolds with application to holomorphic mappings
Recommended article: Invariant holomorphic extension in several complex variables
Recommended article: Q4024771
description: scientific article; zbMATH DE number 1160132