Embeddings of Stein manifolds of dimension \(n\) into the affine space of dimension \(3n/2+1\) (Q1198789)

The existence is proved of proper holomorphic mappings of Stein manifolds of dimension \(n\) into \(\mathbb{C}^ q\) for the minimal \(q>(3n+1)/2\). This minimal number equals \((3n/2)+1=[2n/2]+1\), if \(n\) is even (and gives a sharp results in that case), and it equals \((3n+1)/2=[3n/2]+2\), if \(n\) is odd (and possibly may be improved in that case to \([3n/2]+1)\). The proof bases essentially on some new version of Oka-principle type: The existence of certain continuous sections implies the existence of certain holomorphic sections. This paper gives also some comments to the history of embedding-results, including the authors' first version twenty years ago. However, the time of twenty years brought with this paper besides some small improvement in the result of the first version also a new, quite elegant and short proof. Also some other comments are given, for ex. concerning immersions with \(q>(3n-1)/2\) or for the \(C^ \infty\)- case. In the case of spaces (with singularities) more complicated estimates for embedding-numbers \(q\) appear (however certainly not sharp): The reader may look into the following papers: \textit{K. W. Wiegmann}, Invent. Math. 1, 229-242 (1966; Zbl 0148.320) and \textit{A. Reinhardt} ``Einbettungen, Immersionen und reine Immersionen differenzierbarer Räume in Zahlenräume''. Dissertation, Univ. Bochum (1977).