Complex analytic properties of tubes over locally homogeneous hyperbolic affine manifolds
type: Item
instance of: scholarly article
title: Complex analytic properties of tubes over locally homogeneous hyperbolic affine manifolds
zbMATH Open document ID: 0584.32030
author: Satoru Shimizu
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1985-01-01T00:00:00Z
review text: Since the tangent bundle \(T_ M\) of any real affine manifold M admits a natural complex structure, it is of interest to compare properties of M and \(T_ M\). As an example of this philosophy, it is known that for domains M in \({\mathbb{R}}^ n\) that \(T_ M\) Stein is equivalent to M convex. In the present work affine manifolds M whose universal covering is affinely equivalent to a convex domain \(\Omega\) in \({\mathbb{R}}^ n\) containing no complete straight lines are considered. By means of known functions on a certain convex cone V(\(\Omega)\) associated to \(\Omega\) (e.g. the characteristic function of \textit{E. B. Vinberg} [Trans. Moscow Math. Soc. 1963, 340-403 (1965); translation from Trudy Moskov. mat. Obshch. 12, 303-358 (1963; Zbl 0138.433)] it is shown that one can construct a smooth strictly plurisubharmonic function \(\psi_ M\) on an open subset of \(T_ M\) whose complement is either the support of a divisor or else empty. If M is compact, then \(\psi_ M\) is an exhaustion function and if \(\Omega\) is homogeneous, then \(T_ M\) contains no positive-dimensional compact analytic subsets. With these assumptions it then follows that \(T_ M\) is Stein.
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32M99
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 3935458
zbMATH Keywords: tubes over locally homogeneous hyperbolic affine manifolds
zbMATH Keywords: Stein manifold
zbMATH Keywords: analytic subsets
reviewed by: Bruce Gilligan
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228643
OpenAlex ID: W2129298301
cites work: Q4099744
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DOI: 10.2748/TMJ/1178228643
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description: scientific article; zbMATH DE number 3935458