There Exist Nontrivial Threefolds with Vanishing Hodge Cohomology

TL;DR

This paper investigates the structure of certain three-dimensional algebraic manifolds with vanishing Hodge cohomology, demonstrating the existence of nonaffine, nonproduct examples through deformation analysis and explicit constructions.

Contribution

It introduces new examples of threefolds with vanishing Hodge cohomology that are neither affine nor products, using deformation invariants and explicit surface families.

Findings

Existence of nonaffine, nonproduct threefolds with vanishing Hodge cohomology.

Construction of a family of open surfaces parametrized by -0 with specific Kodaira and D-dimensions.

Identification of deformation invariants for certain open surfaces.

Abstract

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 $X-Y$.