A New Criterion for Affineness

We show that an irreducible quasiprojective variety of dimension defined over an algebraically closed field with characteristic zero is an affine variety if and only if ( ) = 0 and ( ) = 0 for all , , where is any hypersurface with sufficiently large degree. A direct application is that an irreducible quasiprojective variety over is a Stein variety if it satisfies the two vanishing conditions. Here, all sheaves are algebraic. 1. Introduction We work over an algebraically closed field with characteristic zero. Affine varieties are important in algebraic geometry. J.-P. Serre introduced sheaf and cohomology techniques to algebraic geometry and discovered his well-known cohomology criterion ([1], [2, Chapter 2, Theorem 1.1]): a variety (or a Noetherian scheme) is an affine variety if and only if for all coherent sheaves on and all , . Goodman and Hartshorne proved that is an affine variety if and only if contains no complete curves and the dimension of the linear space is bounded for all coherent sheaves on [3]. Let be the completion of . In 1969, Goodman also proved that is affine if and only if after suitable blowing up of the closed subvariety on the boundary , the new boundary is a support of an ample divisor, where is the blowing up with center in ([4], [2, Chapter 2, Theorem 6.1]). For any quasiprojective variety , we may assume that the boundary is the support of an effective divisor with simple normal crossings by blowing up the closed subvariety in . is affine if is ample. So, if we can show the ampleness of , then is affine. There are two important criteria for ampleness according to Nakai-Moishezon and Kleiman ([5], [6, Chapter 1, Section 1.5]). Another sufficient condition is that if？？ contains no complete curves and the linear system is base point free, then is affine [2, Chapter 2, Page 64]. Therefore, we can apply base point free theorem if we know the numerical condition of [6, Chapter 3, Page 75, Theorem 3.3]. Neeman proved that if is a quasicompact Zariski open subset of an affine scheme Spec , then is affine if and only if for all [7]. The significance of Neeman’s theorem is that it is not assumed that the ring is Noetherian. In [8], we show that if a quasiprojective variety is Stein, for all , and has algebraically independent nonconstant regular functions, then is an affine variety. In this note, we give a new criterion for affineness. Theorem 1. An irreducible quasiprojective variety of dimension is an affine variety if and only if for all , , and , where is any hypersurface with sufficiently large degree and . By Cartan’s Theorem B,