Affine Algebraic Varieties

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In this paper, we give new criteria for affineness of a variety defined over C. Our main result is that an irreducible algebraic variety Y (may be singular) of dimension d (d≥ 1) defined over C is an affine variety if and only if Y contains no complete curves, Hⁱ(Y, OY)=0 for all i>0 and the boundary X-Y is support of a big divisor, where X is a projective variety containing Y. We construct three examples to show that a variety is not affine if it only satisfies two conditions among these three conditions. We also give examples to demonstrate the difference between the behavior of the boundary divisor D and the affineness of Y. If Y is an affine variety, then the ring Γ(Y, OY) is noetherian. However, to prove that Y is an affine variety, we do not start from this ring. We explain why we do not need to check the noetherian property of the ring Γ(Y, OY) directly but use the techniques of sheaf and cohomology.