[Paper Review] There Exist Nontrivial Threefolds with Vanishing Hodge Cohomology

Jing Zhang|ArXiv.org|Apr 7, 2005

Algebraic Geometry and Number Theory26 references6 citations

TL;DR

This paper constructs the first known examples of nonaffine, nonproduct threefolds with vanishing Hodge cohomology $ H^i(Y, \Omega^j_Y) = 0 $ for all $ i > 0 $, $ j \geq 0 $, by fibered construction over a punctured affine curve using open surfaces of Mohan Kumar's non-affine types. The key contribution is proving the existence of such threefolds via deformation-invariant cohomology analysis and explicit geometric realization using projective bundles over elliptic fibrations.

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 $X-Y$.

Motivation & Objective

• To prove the existence of nonaffine, nonproduct threefolds with vanishing Hodge cohomology $ H^i(Y, \Omega^j_Y) = 0 $ for all $ i > 0 $, $ j \geq 0 $.
• To analyze the deformation invariance of open surfaces with vanishing Hodge cohomology in the context of threefold fibrations.
• To construct explicit examples of such threefolds by realizing them as total spaces of families of open surfaces parametrized over $ \mathbb{C} - \{0\} $.
• To establish conditions under which the global vanishing of Hodge cohomology holds for threefolds fibered over curves, based on fiberwise cohomological properties.

Proposed method

• Use of deformation theory and cohomological invariance to show that if all fibers of a fibration satisfy vanishing Hodge cohomology, then the total space may also satisfy it under suitable conditions.
• Construction of a rank 2 vector bundle $ E $ on an elliptic surface $ Z \to \mathbb{C} - \{0,1\} $ such that $ E|_{C_t} $ is the unique nonsplit extension of $ \mathcal{O}_{C_t} $ by itself for each fiber $ C_t $.
• Application of the projective bundle construction $ X = \mathbb{P}_Z(E) $, with a canonical section $ D \subset X $, to define the open threefold $ Y = X - D $.
• Leveraging Nakayama’s lemma and the triviality of the Picard group on $ \mathbb{C} - \{0,1\} $ to ensure the existence of a global nonsplit extension via $ H^1(Z, \mathcal{O}_Z) \cong \mathbb{C}[x, 1/x, 1/(x-1)] $.
• Use of Goodman and Hartshorne’s cohomology transfer technique to relate cohomology on open fibers to closed fibers, enabling application of upper semicontinuity theorems.
• Verification of vanishing Hodge cohomology on $ Y $ via the long exact sequence in cohomology and the fact that all fibers satisfy the vanishing condition.

Experimental results

Research questions

• RQ1 Can nonaffine, nonproduct threefolds with vanishing Hodge cohomology exist?
• RQ2 What conditions ensure that the global vanishing of $ H^i(Y, \Omega^j_Y) $ holds when all fibers satisfy the same vanishing?
• RQ3 How can deformation-invariant properties of open surfaces be used to classify threefold fibrations with vanishing Hodge cohomology?
• RQ4 Is it possible to construct such threefolds using families of non-affine open surfaces as fibers?

Key findings

• The paper constructs a nonaffine, nonproduct threefold $ Y $ with $ H^i(Y, \Omega^j_Y) = 0 $ for all $ i > 0 $, $ j \geq 0 $, by realizing it as $ Y = X - D $, where $ X = \mathbb{P}_Z(E) $ and $ D $ is the canonical section of the projective bundle.
• The threefold $ Y $ is fibered over the affine curve $ \mathbb{C} - \{0\} $, with each smooth fiber being an open surface of type (2) in Mohan Kumar’s classification—specifically, the complement of a canonical section in $ \mathbb{P}_C(E) $, where $ E $ is the unique nonsplit extension of $ \mathcal{O}_C $ by itself.
• The construction ensures that $ Y $ has Kodaira dimension $ -\infty $ and $ D $-dimension 1, with $ D $ the boundary divisor supporting $ X - Y $.
• The threefold $ Y $ is not affine, as shown by the fact that its fibers are not affine, and the global vanishing of Hodge cohomology is preserved due to the deformation-invariant cohomological structure of the fibration.
• The example demonstrates that the vanishing of Hodge cohomology does not imply affineness in dimension 3, even when the base is affine and fibers are Stein.
• The paper confirms that the global vanishing of Hodge cohomology on $ Y $ follows from the fiberwise vanishing and the use of upper semicontinuity and cohomological transfer techniques, despite the lack of global ampleness of the boundary divisor.