There Exist Nontrivial Threefolds with Vanishing Hodge Cohomology

Details There Exist Nontrivial Threefolds with Vanishing Hodge Cohomology There Exist Nontrivial Threefolds with Vanishing Hodge Cohomology

2005-04-07

arxiv.org/abs/math/0504142v3

Mathematics

Algebraic Geometry

Revised version. Accepted by Michigan Math. J

Scientific paper

We analyze the structure of the algebraic manifolds $Y$ of dimension 3 with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$, by showing the deformation invariant of some open surfaces. Secondly, we show when a smooth threefold with nonconstant regular functions satisfies the vanishing Hodge cohomology. As an application, we prove the existence of nonaffine and nonproduct threefolds $Y$ with this property by constructing a family of a certain type of open surfaces parametrized by the affine curve $\C-\{0\}$ such that the corresponding smooth completion $X$ has Kodaira dimension $-\infty$ and $D$-dimension 1, where $D$ is the effective boundary divisor with support $X-Y$.

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