Proper holomorphic mappings of positive codimension in several complex variables
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Title: 
Proper holomorphic mappings of positive codimension in several complex variables 
Author(s): 
Chiappari, Stephen Anthony

Doctoral Committee Chair(s): 
Miles, Joseph

Department / Program: 
Mathematics 
Discipline: 
Mathematics 
Degree Granting Institution: 
University of Illinois at UrbanaChampaign 
Degree: 
Ph.D. 
Genre: 
Dissertation 
Subject(s): 
Mathematics

Abstract: 
A holomorphic mapping f from a bounded domain $\Omega$ in C$\sp{\rm n}$ to a bounded domain $\Omega\sp\prime$ in C$\sp{\rm N}$ is proper if and only if (f(z$\sb\nu$)) tends to the boundary b$\Omega\sp\prime$ for each sequence (z$\sb\nu$) that tends to b$\Omega$. If the domains are balls B$\sb{\rm n}$ and B$\sb{\rm N}$, Forstneric has proved that if f is sufficiently smooth up to the sphere bB$\sb{\rm n}$, then it must be rational, and Cima and Suffridge have shown that it then extends to be holomorphic past bB$\sb{\rm n}$. We prove the more general result that if (i) $\Omega$ lies on one side of a real analytic real hypersurface M in C$\sp{\rm n}$, (ii) F maps $\Omega$ holomorphically into the ball B$\sb{\rm N}$, (iii) in some neighborhood of a point p of M, F is the quotient of a holomorphic mapping by a holomorphic function, and (iv) if for each point q of M sufficiently near p, (F(z$\sb\nu$)) tends to bB$\sb{\rm N}$ as (z$\sb\nu$) tends to q within $\Omega$, then F extends to be holomorphic past M at p. We prove this extension result also for certain other target domains, e.g., generalized ellipsoids $\rm\{\sum\sb{j}\ \vert w\sb{j}\vert\sp{2m\sb j}< 1\}$ in $\rm C\sp{N}.$ We also investigate some properties of a certain variety associated to a proper holomorphic mapping and compute it for several mappings that are of importance to the study of proper mappings invariant under fixed point free finite unitary groups and to the classification of polynomial proper mappings between balls. 
Issue Date: 
1990 
Type: 
Text 
Language: 
English 
URI: 
http://hdl.handle.net/2142/19419

Rights Information: 
Copyright 1990 Chiappari, Stephen Anthony 
Date Available in IDEALS: 
20110507 
Identifier in Online Catalog: 
AAI9114202 
OCLC Identifier: 
(UMI)AAI9114202 
Items in IDEALS are protected by copyright, with all rights reserved, unless otherwise indicated.
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