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 $H^i(Y,{\Omega }_Y^j)=0$ for all $j\ge 0$, $i>0$ and $\kappa (D,X)=n$, 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 ${\Omega }_Y^j$ 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].

19 pages

References in corpus (2)

• There Exist Nontrivial Threefolds with Vanishing Hodge Cohomology
• On the $D$-dimension of a certain type of threefolds