[Paper Review] On the $D$-dimension of a certain type of threefolds

Jing Zhang|ArXiv.org|Oct 28, 2006

Algebraic Geometry and Number Theory21 references3 citations

TL;DR

This paper investigates the $D$-dimension of smooth threefolds $Y$ with trivial cohomology $H^i(Y, \Omega^j_Y) = 0$ for all $j \geq 0$, $i > 0$, and shows that the $D$-dimension of any smooth completion $X$ with simple normal crossing boundary divisor $D$ cannot be 2. It further proves that $Y$ is affine if and only if it is regularly separable, and establishes that $Y$ is isomorphic to $\mathrm{Spec}\, \Gamma(Y, \mathcal{O}_Y)$ when $\kappa(D,X) > 1$, unifying cohomological vanishing, regular separability, and affineness in higher dimensions.

ABSTRACT

Let $Y$ be an algebraic manifold of dimension 3 with $H^i(Y, Ω^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$. 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$Γ(Y, {\mathcal{O}}_Y)$. Furthermore, we prove that an algebraic manifold $Y$ of any dimension $d\geq 1$ is affine if and only if $H^i(Y, Ω^j_Y)=0$ 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 $f(y_1) eq f(y_2)$.

Motivation & Objective

• To classify threefolds $Y$ with $H^i(Y, \Omega^j_Y) = 0$ for all $j \geq 0$, $i > 0$, via the $D$-dimension of smooth completions.
• To determine the possible values of the $D$-dimension $\kappa(D,X)$ for such threefolds.
• To establish a characterization of affineness in terms of cohomological vanishing and regular separability for algebraic manifolds of any dimension.
• To generalize results from surfaces to higher-dimensional varieties using fibre space structures and $D$-dimension theory.

Proposed method

• Uses Iitaka's $D$-dimension theory to analyze the growth of global sections of $\mathcal{O}_X(mD)$ for effective divisor $D$ on a smooth completion $X$ of $Y$.
• Applies the fibre space construction from previous work ([Zh1, Zh2]) to reduce the problem to studying the $D$-dimension of general fibres.
• Employs Fujita's result on the $D$-dimension of fibred varieties to rule out $\kappa(D,X) = 2$.
• Uses Zariski’s Main Theorem and Neeman’s theorem to show that an injective birational morphism from $Y$ to $\mathrm{Spec}\, \Gamma(Y, \mathcal{O}_Y)$ implies affineness.
• Applies induction on dimension to extend the affineness criterion to arbitrary-dimensional algebraic manifolds.
• Leverages the upper semi-continuity of $D$-dimension and the connectedness of the boundary to control the global structure of $Y$.

Experimental results

Research questions

• RQ1 Can the $D$-dimension $\kappa(D,X)$ of a smooth completion $X$ of a threefold $Y$ with trivial cohomology be 2?
• RQ2 Under what conditions is such a threefold $Y$ affine, given its cohomological vanishing and regular separability?
• RQ3 Is the scheme associated to $Y$ isomorphic to $\mathrm{Spec}\, \Gamma(Y, \mathcal{O}_Y)$ when $\kappa(D,X) > 1$?
• RQ4 Does regular separability imply affineness for algebraic manifolds satisfying the cohomological vanishing condition?
• RQ5 What is the relationship between $\kappa(D,X)$, the absence of complete curves in $Y$, and the affineness of $Y$?

Key findings

• The $D$-dimension $\kappa(D,X)$ of any smooth completion $X$ of a threefold $Y$ with $H^i(Y, \Omega^j_Y) = 0$ for all $j \geq 0$, $i > 0$ cannot be 2.
• If $\kappa(D,X) > 1$, then $Y$ is isomorphic to $\mathrm{Spec}\, \Gamma(Y, \mathcal{O}_Y)$, meaning $Y$ is affine if and only if it is regularly separable.
• For any dimension $d \geq 1$, an algebraic manifold $Y$ is affine if and only if $H^i(Y, \Omega^j_Y) = 0$ for all $j \geq 0$, $i > 0$, and $Y$ is regularly separable.
• When $\kappa(D,X) > 1$, the $D$-dimension of $X$ is exactly 3, and $Y$ is affine by the combination of birational injectivity and Neeman’s theorem.
• The $D$-dimension of $X$ is 3 if and only if all general fibres of the fibre space structure on $Y$ are affine surfaces with $D$-dimension 2.
• If $Y$ is not affine and satisfies the cohomological vanishing, then $\kappa(X) = -\infty$ and $\kappa(D,X) = 1$ for every smooth completion $X$.