Fibre cycles of holomorphic maps. II: Fibre cycle space and canonical flattening
type: Item
instance of: scholarly article
title: Fibre cycles of holomorphic maps. II: Fibre cycle space and canonical flattening
zbMATH Open document ID: 0832.32007
DOI: 10.1007/BF01450486
author: Bernd Siebert
published in: Mathematische Annalen
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/165252
review text: The author concludes the study of the space \(Z(F)\) of limits of generic fibres of a not necessarily proper holomorphic map \(F : Y \to X\) [see part I, ibid. 296, No. 2, 269-283 (1993; Zbl 0807.32011)]. He considers geometric flatness which, for a proper surjective holomorphic map \(F : Y \to X\) of constant fibre dimension means that there is a holomorphic map \(Z_F : X \to C_d (Y)\) where \(|Z_F(x) |= F^{-1}(x)\) for all \(x\) and \(C_d (Y)\) is the space of compact analytic cycles. When \(F\) is not proper but generically open and \(X\) is seminormal, this notion can be generalised by considering generic fibres. Geometric flatness is weaker than algebraic flatness. The main result is that, under some finiteness condition on \(F\), \(Z(F)\) is a complex space and has the universal geometric flattening property. This means the following. Consider the strict transform of \(F\) under the natural map \(Z(F) \to X\); it is a geometrically flat map and \(Z(F)\) is minimal with respect to this property.
Mathematics Subject Classification ID: 32C15
Mathematics Subject Classification ID: 32H99
zbMATH DE Number: 704905
zbMATH Keywords: fiber cycle
zbMATH Keywords: canonical flattening
zbMATH Keywords: holomorphic map
reviewed by: Fabrizio Broglia
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 704905