Embeddings of Stein spaces into affine spaces of minimal dimension
type: Item
instance of: scholarly article
title: Embeddings of Stein spaces into affine spaces of minimal dimension
zbMATH Open document ID: 0881.32007
published in: Mathematische Annalen
publication date: 1997-10-23T00:00:00Z
review text: Let \(Y\) be a finite-dimensional Stein manifold and \((X,{\mathcal O}_X)\) an \(n\)-dimensional closed analytic subspace of bounded embedding dimension \(m:= \text{embdim} (X,{\mathcal O}_X).\) Let \(n(k)(k\in \mathbb{N}_0)\) denote the complex dimension of the analytic set \(\{x\in X: \text{embdim}_x (X,{\mathcal O}_X)\geq k \}\) and let \[ b' (X,{\mathcal O}_X) := \sup \{ k+\left[\frac{n(k)}{2}\right ]: k \in \mathbb{N}_0, k \leq m \} . \] As an answer to the question, how far the dimension \(N\) of the embedding theorem of \(X\) into the affine space \(\mathbb{C}^N\) can be lowered, the author proves the theorem: there exists a holomorphic map \(f:Y \to\mathbb{C}^N \) with \(N = \max \{ n+ \left[\frac{n}{2}\right ] +1, b' (X,{\mathcal O}_X),3 \}\) such that the analytic restriction of \(f\) induces an embedding of \((X,{\mathcal O}_X)\).
Mathematics Subject Classification ID: 32E10
zbMATH DE Number: 1031538
zbMATH Keywords: Stein spaces
zbMATH Keywords: Stein manifolds
zbMATH Keywords: embedding dimension
zbMATH Keywords: minimal dimension
author: Jörg Schürmann
reviewed by: S. G. Samko
MaRDI profile type: Publication
full work available at URL: s002080050040
OpenAlex ID: W2094506437
DOI: 10.1007/S002080050040
description: scientific article; zbMATH DE number 1031538