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 $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ 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 $\Omega^j_Y$ 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].
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- October 2006
- DOI:
- 10.48550/arXiv.math/0610884
- arXiv:
- arXiv:math/0610884
- Bibcode:
- 2006math.....10884Z
- Keywords:
-
- Mathematics - Algebraic Geometry;
- Mathematics - Complex Variables;
- 14J10;
- 14J30;
- 32E10
- E-Print:
- 19 pages