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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 $$d\geq 1$$.

##### MSC:
 32E10 Stein spaces, Stein manifolds 14J10 Families, moduli, classification: algebraic theory 32Q28 Stein manifolds 14J40 $$n$$-folds ($$n>4$$)
##### Keywords:
Stein variety; affine variety; quasi-projective variety
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