Fibre cycles of holomorphic maps. I: Local flattening
type: Item
instance of: scholarly article
title: Fibre cycles of holomorphic maps. I: Local flattening
zbMATH Open document ID: 0807.32011
DOI: 10.1007/BF01445106
author: Bernd Siebert
published in: Mathematische Annalen
publication date: 1995-02-19T00:00:00Z
full work available at URL: https://eudml.org/doc/165086
review text: The author studies the space \(Z(F)\) of limits of sufficiently general sequences \(\{F^{-1} (x_ \nu)\}_ \nu\) for a generically open holomorphic map; to this space can be given a complex structure in a natural way. In this first paper, as an application, he proves the following local geometric flattening theorem: Theorem: Let \(F:Y \to X\) be a generically open holomorphic map and \(K \subset Y\) compact. Then there exists a holomorphic map \(\sigma : X' \to X\), \(X'\) a normal complex space, and a relatively compact, open neighbourhood \(C\) of \(K\), such that the following holds: Let \(Y' \subset X' \times_ XC\) consist of those irreducible components of \(X' \times_ X C\) projecting generically open to \(X'\). Then the projection to the first factor \(F' : Y' \to X'\) is open and surjective. The projection to the second factor \(F' : Y' \to C\) is semiproper and surjective, \(\sigma\) is generically open and generically finite. As the author remarks, since the map \(\pi\) in the theorem is always semiproper, one can derive the result as a consequence of the local flattening theorem proved by Hironaka et al. Nevertheless the study of \(Z(F)\) is interesting by itself and gives a more elementary, fully geometric proof.
Mathematics Subject Classification ID: 32C15
Mathematics Subject Classification ID: 32H99
Mathematics Subject Classification ID: 32C20
Mathematics Subject Classification ID: 32C25
zbMATH DE Number: 537298
zbMATH Keywords: fiber cycles
zbMATH Keywords: local flattening
reviewed by: Fabrizio Broglia
MaRDI profile type: Publication
cites work: Complex analytic geometry
cites work: Points de platitude d'un morphisme d'espaces analytiques complexes
cites work: Komplexe Räume
cites work: Q3322387
cites work: Q3720931
cites work: Flattening Theorem in Complex-Analytic Geometry
cites work: Q4773815
cites work: Zur Konstruktion komplexer Basen
cites work: Q4052252
OpenAlex ID: W1508486465
description: scientific article; zbMATH DE number 537298