Hodge Cohomology Criteria For Affine Varieties

Abstract

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 $H^i(Y, Ω^j_Y)=0$ for all $j\geq 0$, $i>0$ 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 $Ω^j_Y$ 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 \cite{36} in algebraic case since the associated analytic space of an affine variety is Stein [15, Chapter VI, Proposition 3.1].