On the $D$-dimension of a certain type of threefolds

TL;DR

This paper investigates the properties of certain threefolds with vanishing cohomology, establishing restrictions on their $D$-dimension and characterizing when such manifolds are affine based on cohomological and separability conditions.

Contribution

It proves that the $D$-dimension of these threefolds cannot be 2, relates the $D$-dimension to the scheme structure, and characterizes affine manifolds via cohomology and regular separability.

Findings

The $D$-dimension of the threefolds cannot be 2.

If the $D$-dimension exceeds 1, the scheme of $Y$ is isomorphic to its global sections.

A manifold is affine iff it has vanishing cohomology and is regularly separable.

Abstract

Let $Y$ be an algebraic manifold of dimension 3 with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$. Let $X$ be a smooth completion of $Y$ such that the boundary $X-Y$ is the support of an effective divisor $D$ on $X$ with simple normal crossings. We prove that the $D$-dimension of $X$ cannot be 2, i.e., either any two nonconstant regular functions are algebraically dependent or there are three algebraically independent nonconstant regular functions on $Y$. Secondly, if the $D$-dimension of $X$ is greater than 1, then the associated scheme of $Y$ is isomorphic to Spec$\Gamma(Y, {\mathcal{O}}_Y)$. Furthermore, we prove that an algebraic manifold $Y$ of any dimension $d\geq 1$ is affine if and only if $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and it is regularly separable, i.e., for any two distinct points $y_1$, $y_2$ on $Y$, there is a regular function $f$ on $Y$ such that $f(y_1)\neq f(y_2)$.