Interpolation of embeddings of Stein manifolds on discrete sets
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.
scientific article; zbMATH DE number 1982571
scientific article