Interpolation of embeddings of Stein manifolds on discrete sets
type: Item
instance of: scholarly article
title: Interpolation of embeddings of Stein manifolds on discrete sets
zbMATH Open document ID: 1037.32024
author: Jasna Prezelj
published in: Mathematische Annalen
publication date: 2003-09-16T00:00:00Z
review text: One of the main results of this paper can be stated as follows: Theorem 1.1 (a): Let \(X\) be an \(n\)-dimensional Stein manifold, \(Y\) a discrete subset of \(X\), \(N:= \max\{[{n+1\over 2}]+ 1,3\}\) and \(\varphi: Y\to \mathbb{C}^{n+q}\) for some \(q\geq 0\). Assume that \(\varphi\) is proper and injective. If (\(n= 1\) and \(q\geq 2\)) or (\(n\geq 1\) and \(q\geq N\)), then there exists a proper holomorphic embedding \(\Phi: X\to\mathbb{C}^{n+q}\) extending \(\varphi\). This result is an analogue of the Schürmann embedding theorem with additional interpolation on discrete sets which is optimal for even \(n\). On the other hand, as far as interpolation on arbitrary closed submanifold \(Y\) in \(X\) is concerned, the author provides a counterexample to the following result of \textit{F. Acquistapace}, \textit{F. Broglia} and \textit{A. Tognoli} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 2, 507--522 (1975; Zbl 0313.32020)]. Theorem: Let \(X\) be a Stein manifold with \(\dim X=n\) and \(Y\subset X\) a closed complex submanifold. Then any proper holomorphic embedding \(f: Y\to\mathbb{C}^m\) for \(m\geq 2n+1\) can be extended to a proper holomorphic embedding \(F: X\to\mathbb{C}^m\) if \(m\) in the above theorem is \(\leq 2n-1\). For \(m= 2n\), the problem is still open. The paper is very well written.
reviewed by: Tan Vovan
Mathematics Subject Classification ID: 32Q40
Mathematics Subject Classification ID: 32Q28
zbMATH DE Number: 1982571
zbMATH Keywords: Stein manifold
zbMATH Keywords: proper holomorphic embedding
zbMATH Keywords: interpolation of embeddings
MaRDI profile type: Publication
full work available at URL: s00208-003-0401-5
OpenAlex ID: W1983946647
DOI: 10.1007/S00208-003-0401-5
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description: scientific article; zbMATH DE number 1982571