Steins, affines and Hilbert’s fourteenth problem. (English) Zbl 0685.14002

Neeman, Ammon

Ann. Math. (2) 127, No. 2, 229-244 (1988).

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

Keywords:

quasi-affine schemes; Stein spaces; affine normal variety; Hilbert’s 14th problem