A New Criterion for Affineness. (20th March 2013) - Zhang, Jing
Abstract : We show that an irreducible quasiprojective variety Y of dimension d ≥ 1 defined over an algebraically closed field with characteristic zero is an affine variety if and only if H i (Y,  Y ) = 0 and H i (Y,  Y ( − Z ) ) = 0 for all i > 0, Z = H ∩ Y, where H is any hypersurface with sufficiently large degree. A direct application is that an irreducible quasiprojective variety Y over ℂ is a Stein variety if it satisfies the two vanishing conditions. Here, all sheaves are algebraic.