There Exist Nontrivial Threefolds with Vanishing Hodge Cohomology

We analyze the structure of the algebraic manifolds $Y$ of dimension 3 with $H^i(Y,{\mathrm{\Omega }}_Y^j)=0$ for all $j\ge 0$, $i>0$ 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 $\text{C}-\{0\}$ such that the corresponding smooth completion $X$ has Kodaira dimension $-\mathrm{\infty }$ and $D$-dimension 1, where $D$ is the effective boundary divisor with support $X-Y$.