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Algebraic Stein varieties. (English) Zbl 1160.32014
An affine complex variety is Stein, i.e., holomorphically separable and holomorphically convex, but there are quasi-projective Stein surfaces without non-constant regular functions. There are also quasi-affine Stein varieties which are not affine.
In the present paper, it is shown that a \(d\)-dimensional irreducible quasi-projective Stein variety \(Y\) is affine iff the transcendence degree \(t\) of \(\Gamma(Y,{\mathcal O}_Y)\) over \(\mathbb C\) equals \(d\) and the cohomology groups \(H^i(Y,{\mathcal O}_Y)\) vanish for all \(i>0\), where \({\mathcal O}_Y\) denotes the sheaf of regular functions on \(Y\). From the maximality condition of the transcendence degree it follows that for every closed curve \(C\) in \(Y\) there exists \(f\in \Gamma(Y,{\mathcal O}_Y)\) which is not constant on \(C\). Therefore \(Y\) is quasi-affine by a result of J. Goodman and R. Hartshorne [Am. J. Math. 91, 258–266 (1969; Zbl 0176.18303)], and the cohomology condition then implies that \(Y\) is affine by a theorem of A. Neeman [Ann. Math. (2) 127, No. 2, 229–244 (1988; Zbl 0685.14002)]. The author gives a proof by induction on \(d\), starting with a detailed description of the surface case, where the cohomology condition is not needed. She also shows that \(d-t\in 2\mathbb N\) for all \(d\geq 1\).

32E10 Stein spaces, Stein manifolds
14J10 Families, moduli, classification: algebraic theory
32Q28 Stein manifolds
14J40 \(n\)-folds (\(n>4\))
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