A generalisation of a theorem of Fornaess-Sibony
type: Item
instance of: scholarly article
title: A generalisation of a theorem of Fornaess-Sibony
zbMATH Open document ID: 0584.32045
DOI: 10.1007/BF01455918
author: Mechthild Behrens
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164042
review text: On any complex manifold M one may consider those tangent vectors \(\xi\) with \(F_ M(p,\xi)=0\), where \(F_ M\) denotes the infinitesimal Kobayashi metric. In general settings it is difficult to say very much about this because one can find examples of manifolds M such that \(\{\xi | F_ M(p,\xi)=0\}\) is a vector subspace of the tangent space whose dimension \((=:\) the corank of \(F_ M)\) depends on the point \(p\in M.\) One possibility is to assume M is exhausted by biholomorphic images of a fixed manifold \(\Omega\). If \(\Omega\) is hyperbolic and \(\Omega\) /Aut(\(\Omega)\) is compact, \textit{J. E. Fornaess} and \textit{N. Sibony} [Math. Ann. 255, 351-360 (1981; Zbl 0438.32012)] showed that the corank of \(F_ M\) is independent of p. Further, if this corank is one, then they proved that M is biholomorphic to a locally trivial holomorphic fiber bundle with fiber \({\mathbb{C}}\). In the present work the author considers manifolds M exhausted by biholomorphic images of a fixed strongly pseudoconvex \(C^ 2\) domain \(\Omega \subset \subset {\mathbb{C}}^ n\) and proves analogous results in this set-up. The proof uses the fact that such \(\Omega\) are taut and an argument of \textit{A. Kodama} [''On complex manifolds exhausted by biholomorphic images of a strictly pseudoconvex domain'', Preprint 1984] is needed to handle the case where a subsequence of maps converges to a constant map into the boundary of \(\Omega\). \(\{\) The reviewer has some results in the homogeneous setting (''On the Kobayashi pseudometric reduction of solv-manifolds'', to appear)\(\}\).
Mathematics Subject Classification ID: 32F45
Mathematics Subject Classification ID: 32L05
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 3935473
zbMATH Keywords: exhaustion of complex manifolds by biholomorphic images of a strongly pseudoconvex domain
zbMATH Keywords: Kobayashi metric
zbMATH Keywords: holomorphic fiber bundle
reviewed by: Bruce Gilligan
MaRDI profile type: Publication
cites work: Biholomorphic Invariants of a Hyperbolic Manifold and some Applications
cites work: Q3315776
cites work: Q3855454
cites work: Q5723628
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description: scientific article; zbMATH DE number 3935473