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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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