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Steins, affines and Hilbert’s fourteenth problem. (English) Zbl 0685.14002
Let U be Zariski open in X, an affine normal variety of finite type over $${\mathbb{C}}$$. Let $$U^{an}$$ be the analytic form of U. This paper shows that:
(1) if X is smooth, then $$U^{an}$$ is Stein iff U is affine,
(2) if $$U^{an}$$ is Stein, and if $$\Gamma$$ (U,$${\mathcal O}_ U)$$ is a finitely generated $${\mathbb{C}}$$-algebra, then U is affine,
(3) the condition on $$\Gamma$$ (U,$${\mathcal O}_ U)$$ in (2) cannot be removed. For this, a 3-dimensional quasi-affine, not affine U such that $$U^{an}$$ is Stein is constructed. Hence: $$\Gamma$$ (U,$${\mathcal O}_ U)$$ is not finitely generated, after (2).
This gives a Stein counterexample to Zariski’s version of Hilbert’s 14th problem.
Reviewer: F.Campana

MSC:
 14A15 Schemes and morphisms 32E10 Stein spaces, Stein manifolds
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