Hodge Cohomology Criteria for Affine Varieties

We give several new criteria for a quasi-projective variety to be affine. In particular, we prove that an algebraic manifold Y with dimension n is affine if and only if H(Y,ΩjY ) = 0 for all j ≥ 0, i > 0 and κ(D,X) = n, i.e., there are n algebraically independent nonconstant regular functions on Y , where X is the smooth completion of Y , D is the effective boundary divisor with support X−Y and ΩjY is the sheaf of regular j-forms on Y . This proves Mohan Kumar’s affineness conjecture for algebraic manifolds and gives a partial answer to J.P. Serre’s Steinness question [36] in algebraic case since the associated analytic space of an affine variety is Stein [15, Chapter VI, Proposition 3.1]. Department of Mathematics, University of Missouri, Columbia, MO 65211, USA Email address: [email protected] 2000 Mathematics Subject Classification: 14J10, 14J30, 32E10.