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A new criterion for affineness. (English) Zbl 1267.14003
The author works over an algebraically closed field \(k\) of characteristic \(0\). He shows that an irreducible quasiprojective variety \(Y\) of dimension \(d \geq 1\) defined over \(k\) is affine if and only if \(H^i(Y, \mathcal{O}_Y) = 0\) and \(H^i(Y, \mathcal{O}_Y(-H \cap Y)) = 0\) for all \(i > 0\), where \(H\) is a hypersurface with sufficiently large degree. In the proof of this theorem the author uses J.-P. Serre’s criterion for affineness [J. Math. Pures Appl., IX. Sér. 36, 1–16 (1957; Zbl 0078.34604)], J. Goodman and R. Hartshorne’s theorem [Am. J. Math. 91, 258–266 (1969; Zbl 0176.18303)] and A. Neeman’s result [Ann. Math. (2) 127, No. 2, 229–244 (1988; Zbl 0685.14002)]. As an application the author applies his criterion for affineness to Stein varieties.
MSC:
14A15 Schemes and morphisms
32E10 Stein spaces
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