The structure of rational homogeneous domains in \({\mathbb{C}}^ n\)
type: Item
instance of: scholarly article
title: The structure of rational homogeneous domains in \({\mathbb{C}}^ n\)
zbMATH Open document ID: 0655.32027
DOI: 10.2307/1971404
published in: Annals of Mathematics. Second Series
publication date: 1987-01-01T00:00:00Z
review text: Bounded homogeneous domains in \({\mathbb{C}}^ n\) have been investigated from many different angles by numerous authors since Eli Cartan started the subject in 1935. One of the highlights in this field is the work of Piatetski-Shapiro who showed that bounded homogeneous domains are biholomorphically equivalent with homogeneous ''Siegel domains''. A simple consequence of his work is that such a Siegel domain \(\Omega\) is (i) contractible, (ii) homogeneous under a connected Lie group G, (iii) ``rational'', i.e., for all \(g\in G\) the map \(z\to g\cdot z\) of \(\Omega\) onto \(\Omega\) is rational and (iv) G acts effectively. The purpose of this paper is to study ``rational, homogeneous domains'', i.e., domains in \({\mathbb{C}}^ n\) satisfying the four properties above. The first main result of this paper is Theorem (0.1). Let \(\Omega\) be a domain in \({\mathbb{C}}^ n\). Let G be a connected Lie group which acts rationally on \(\Omega\). Then there exists a real, algebraic group \(G_ R\) such that (i) G is a Zariski-dense, analytic subgroup of \(G_ R.\) (ii) The action \(\mu\) : \(G\times \Omega \to \Omega\) extends to a rational mapping \({\bar \mu}\): \(G_ R\times {\mathbb{C}}^ n\to {\mathbb{C}}^ n\) (which may have singularities). Furthermore, if G acts transitively on \(\Omega\), then \(G_ R\) may be chosen so that \(\Omega\) is homogeneous under the connected component \((G_ R)_ 0.\) The second main result of the paper is a description of the rationally homogeneous domains in \({\mathbb{C}}^ n\) as ``Siegel domains of type N-P''. ``To define this concept, let N be a connected, simply connected nilpotent Lie group with Lie algebra \({\mathcal N}\). Let \({\mathcal N}_ c\) be the complexification of \({\mathcal N}\) and let \(N_ c\) be the corresponding complex Lie group. Let \(P\subset N_ c\) be a complex, connected subgroup. Any automorphism of N extends holomorphically to an automorphism of \(N_ c\). Let \(A_ p\) be the group of automorphisms of N which lea…
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 22E10
zbMATH DE Number: 4069316
zbMATH Keywords: rationally homogeneous domains in \({\mathbb{C}}^ n\)
zbMATH Keywords: Siegel domains of type N-P
author: Richard Penney
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971404
OpenAlex ID: W2317681103
description: scientific article; zbMATH DE number 4069316