Proper holomorphic mappings of positive codimension in several complex variables
Chiappari, Stephen Anthony
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https://hdl.handle.net/2142/19419
Description
Title
Proper holomorphic mappings of positive codimension in several complex variables
Author(s)
Chiappari, Stephen Anthony
Issue Date
1990
Doctoral Committee Chair(s)
Miles, Joseph B.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
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