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Embeddings of Stein spaces into affine spaces of minimal dimension. (English) Zbl 0881.32007
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 $$(X,{\mathcal O}_X)$$.
Reviewer: S.G.Samko (Faro)

##### MSC:
 3.2e+11 Stein spaces
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