Embeddings of Stein spaces into affine spaces of minimal dimension
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 \((X,{\mathcal O}_X)\).
scientific article; zbMATH DE number 1031538