There Exist Nontrivial Threefolds with Vanishing Hodge Cohomology

Abstract

We analyze the structure of the algebraic manifolds $Y$ of dimension 3 with $H^i(Y, Ω^j_Y)=0$ 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 $X-Y$.