On the number of factors in the unipotent factorization of holomorphic mappings into \(\mathrm{SL}_{2}(\mathbb {C})\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1250.32009
DOI: 10.1090/S0002-9939-2011-11025-6
author: Björn Ivarsson
author: Frank Kutzschebauch
published in: Proceedings of the American Mathematical Society
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.1025
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 6024090
zbMATH Keywords: Stein space
zbMATH Keywords: holomorphic submersion
zbMATH Keywords: spray
MaRDI profile type: Publication
arXiv ID: 1012.1025
cites work: The Oka principle for sections of stratified fiber bundles
cites work: Extending holomorphic sections from complex subvarieties
cites work: Oka's principle for holomorphic submersions with sprays
cites work: Oka's Principle for Holomorphic Sections of Elliptic Bundles
cites work: A solution of Gromov's Vaserstein problem
cites work: On Kazhdan's property (T) for the special linear group of holomorphic functions
cites work: Q4124941
cites work: Reduction of a Matrix Depending on Parameters to a Diagonal Form by Addition Operations
cites work: Q3836619
OpenAlex ID: W1568518243
Recommended article: A solution of Gromov's Vaserstein problem
Recommended article: Holomorphic factorization of mappings into \(\mathrm{SL}_n(\mathbb{C})\)
Recommended article: On Kazhdan's property (T) for the special linear group of holomorphic functions
Recommended article: Holomorphic factorization of mappings into \(\mathrm{Sp}_4 (\mathbb{C})\)
Recommended article: Unipotent factorization of vector bundle automorphisms
Recommended article: Connected components of regular fibers of differentiable maps
Recommended article: A CHARACTERIZATION OF Ck×(C*)ℓFROM THE VIEWPOINT OF BIHOLOMORPHIC AUTOMORPHISM GROUPS
Recommended article: Q4697154
Recommended article: On the holomorphic separability of discrete quotients of complex Lie groups
Recommended article: Q3798956
title: On the number of factors in the unipotent factorization of holomorphic mappings into \(\mathrm{SL}_{2}(\mathbb {C})\)
review text: The starting point is Vaserstein's theorem: For any natural number \(n\) and integer \(d \geq 0\) there is a natural number \(K\) such that for any finite dimensional normal topological space \(X\) of dimension \(d\) and a null-homotopic, that is homotopic to the identity, continuous mapping \(f : X \rightarrow \mathrm{SL}_{n}(\mathbb {C})\), \(f\) can be written as a finite product of no more than \(K\) unipotent matrices. That is, one can find continuous mappings \(F_l : X \rightarrow \mathbb {C}^{n(n-1)/2}\), \(1 \geq l \geq K\), such that \(f(x) = M_1\big(F_1(x)\big)\dots M_K\big(F_K(x)\big)\), where \(M_j(F_j)\) are upper or lower triangular idempotent matrices.NEWLINENEWLINEIn their previous paper [the authors, Ann. Math. (2) 175, No. 1, 45--69 (2012; Zbl 1243.32007)] the authors have proved that if \(X\) is a \(d\)-dimensional Stein space then there exists a number \(K'>K\) depending on \(d\) and \(n\) such that there exists such a factorization by holomorphic upper or lower triangular idempotent matrices. The main tool in the proof is the homotopy principle and the construction of a suitable submersion with a spray. However, due to the choice of the fibration, the number of factors needed in the holomorphic case is much larger than in the continuous one. Therefore the authors reprove their factorization theorem in the case of \(SL_2(\mathbb{C})\) by using a different submersion. As a result they get \(K' = K + 2\) and prove that \(K' = 4\) if \(n = 1\) and \(K' = 5\) if \(n = 2\).
reviewed by: Jasna Prezelj
description: scientific article; zbMATH DE number 6024090