A New Criterion for Affineness

Abstract

We show that an irreducible quasiprojective variety of dimension ≥ 1 defined over an algebraically closed field with characteristic zero is an affine variety if and only if ( , O ) = 0 and ( , O (− )) = 0 for all > 0, = ∩ , where is any hypersurface with sufficiently large degree. A direct application is that an irreducible quasiprojective variety over C is a Stein variety if it satisfies the two vanishing conditions. Here, all sheaves are algebraic.