A new criterion for affineness (Q1952646)

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 \textit{J.-P. Serre}'s criterion for affineness [J. Math. Pures Appl., IX. Sér. 36, 1--16 (1957; Zbl 0078.34604)], \textit{J. Goodman} and \textit{R. Hartshorne}'s theorem [Am. J. Math. 91, 258--266 (1969; Zbl 0176.18303)] and \textit{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.