Über exzeptionelle Mengen
type: Item
instance of: scholarly article
title: Über exzeptionelle Mengen
zbMATH Open document ID: 0546.32004
DOI: 10.1007/BF01239942
author: Thomas Peternell
published in: Manuscripta Mathematica
publication date: 1982-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/154825
review text: A linear fibre space V over a complex space X is said to be weakly negative if the zero section is exceptional (i.e., can be contracted to a point). If Y is a subspace of X, the normal bundle to Y is the linear (fibre) space \(N_ Y\) dual to the sheaf \(I/I^ 2\), I being the ideal sheaf of Y. \textit{H. Grauert} [Math. Ann. 146, 331-368 (1962; Zbl 0173.330)] proved a famous theorem to the effect that Y is exceptional if \(N_ Y\) is weakly negative. In this paper (and subsequent correction) it is proved, conversely, that if Y' is an exceptional set, then there is a subspace Y, with \(Y_{red}=Y'\), having weakly negative normal bundle. As a corollary, the author proves the converse of a result of \textit{V. Ancona} [Trans. Am. Math. Soc. 215, 45-61 (1976; Zbl 0319.32015)] stating that X is a Moishezon space if there is a weakly negative linear fibre space V on X with all fibres of dimension \(>0\). - Relative versions of the main theorem [which was also proved by \textit{L. Ancona} and \textit{G. Tomassini}, ''Modifications analytiques'', Lect. Notes Math. 943 (1982; Zbl 0498.32006), independently] are given. The proof of theorem 3 is incomprehensible to the reviewer.
Mathematics Subject Classification ID: 32C25
Mathematics Subject Classification ID: 32C15
Mathematics Subject Classification ID: 32J99
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 3869716
zbMATH Keywords: exceptional sets
zbMATH Keywords: Grauert conjecture
zbMATH Keywords: weakly negative fibre space
zbMATH Keywords: weakly negative normal bundle
zbMATH Keywords: characterization of Moishezon space
zbMATH Keywords: complex space
MaRDI profile type: Publication
cites work: Embedding Moishezon spaces into 1-convex spaces
cites work: Über Modifikationen und exzeptionelle analytische Mengen
cites work: Vektorraumbündel in der Nähe von kompakten komplexen Unterräumen
cites work: Familien negativer Vektorraumbündel und 1-konvexe Abbildungen
cites work: Relativexzeptionelle analytische Mengen. (Relative exceptional analytical sets.)
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description: scientific article; zbMATH DE number 3869716