Homogeneous Koszul manifolds in \(\mathbb{C}^ n\)
type: Item
instance of: scholarly article
title: Homogeneous Koszul manifolds in \(\mathbb{C}^ n\)
zbMATH Open document ID: 0770.32017
published in: Journal of Differential Geometry
publication date: 1993-04-01T00:00:00Z
review text: Suppose \(X\) is a complex manifold which is homogeneous under the real Lie group \(G\) acting by holomorphic automorphisms and assume \(X\) has a \(G\)- invariant volume form. The theory of such manifolds was developed by \textit{J. L. Koszul} [Can. J. Math. 7, 562-576 (1955; Zbl 0066.161)] and the case when the Koszul structure is Kähler has been studied in detail, see the work of \textit{M. Dorfmeister} and \textit{K. Nakajima} [Acta Math. 161, No. 1/2, 23-70 (1988; Zbl 0662.32025)] and the references given there. In this paper homogeneous pseudo-Kähler manifolds for which the group contains an exponential solvable group acting transitively on the given space are studied. Such manifolds are referred to as type \(E\). Using a theorem of \textit{J.-I. Hano} [Scripta Math. 29, 125-139 (1971; Zbl 0259.32009)] it is proved that all type \(E\) manifolds may be realized as Siegel \(N - P\) domains, a class of homogeneous domains in \(\mathbb{C}^ n\) introduced by the author [Ann. Math., II. Ser. 126, 389-414 (1987; Zbl 0655.32027)], and that the identity component of the holomorphic isometry group has a rational action. A notion of rank is defined, is shown to be independent of the group acting and attention is then restricted to manifolds of type \(E\) and rank one which are called type I. Koszul domains of type I are shown to be biholomorphic to homogeneous nil-balls. It is conjectured that a contractable homogeneous domain with analytic, Levi nondegenerate boundary is a type \(I\) domain. As a step towards proving a result of this kind it is proved that a contractable domain in \(\mathbb{C}^ n\) which is homogeneous under a rationally homogeneous action of \(G\) and whose boundary is real analytic, codimension-one, Levi nondegenerate, homogeneous under an analytic extension of the action of \(G\) and has an attractive fixed point is type \(I\). By means of a result of \textit{S. M. Webster} [Invent. Math. 43, 53-68 (1977; Zbl 0348.32005)] it is shown that for a domai…
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 146366
zbMATH Keywords: Koszul manifolds
zbMATH Keywords: \(N-P\) Siegel domains
Wikidata QID: Q115180507
author: Richard Penney
reviewed by: Bruce Gilligan
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214453182
OpenAlex ID: W1590252386
DOI: 10.4310/JDG/1214453182
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description: scientific article; zbMATH DE number 146366