[Paper Review] The extended locus of Hodge classes

Christian Schnell|arXiv (Cornell University)|Jan 28, 2014

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper introduces the extended locus of Hodge classes—a finite, proper analytic space that compactifies the usual locus of Hodge classes by including integral classes becoming Hodge in the limit via mixed Hodge structures. Using Saito’s theory of mixed Hodge modules and a generalization of Cattani-Deligne-Kaplan’s theorem, it constructs this space for polarized variations of integral Hodge structure of weight zero, proving holomorphic convexity and showing that all positive-dimensional compact analytic subsets lie within the extended locus.

ABSTRACT

We introduce an "extended locus of Hodge classes" that also takes into account integral classes that become Hodge classes "in the limit". More precisely, given a polarized variation of integral Hodge structure of weight zero on a Zariski-open subset of a complex manifold, we construct a canonical analytic space that parametrizes limits of integral classes; the extended locus of Hodge classes is an analytic subspace that contains the usual locus of Hodge classes, but is finite and proper over the base manifold. The construction uses Saito's theory of mixed Hodge modules and a small generalization of the main technical result of Cattani, Deligne, and Kaplan. We study the properties of the resulting analytic space in the case of the family of hyperplane sections of an odd-dimensional smooth projective variety.

Motivation & Objective

• To address the compactification of Hodge loci in families of hyperplane sections, particularly when Hodge loci have positive dimension or degenerate in the limit.
• To resolve the issue that standard closure in projective space fails to capture limit phenomena such as vanishing cycles becoming Hodge classes in degenerate fibers.
• To define and construct a canonical analytic space that parametrizes integral classes becoming Hodge classes in the limit, extending the classical locus of Hodge classes.
• To prove that this extended locus is finite and proper over the base, and that all positive-dimensional compact analytic subsets lie within it.
• To establish holomorphic convexity of the extended parameter space, using the ampleness of certain cohomological bundles.

Proposed method

• Uses Saito’s theory of mixed Hodge modules to construct a filtered $\mathscr{D}$-module $ (\mathcal{M}, F_\bullet) $ associated to the variation of Hodge structure.
• Applies a generalized version of the Cattani-Deligne-Kaplan theorem to relate the Hodge filtration to the geometry of the parameter space.
• Constructs the extended parameter space $ \tilde{T}_{\mathbb{Z}}(K) $ as a finite holomorphic cover of the total space of the dual of a globally generated vector bundle.
• Utilizes residue maps from meromorphic forms on $ B \times X $ with poles along the universal hyperplane section to realize sections of $ \mathcal{H} $.
• Shows that $ F_{-1}\mathcal{M} $ is a quotient of an ample vector bundle, implying holomorphic convexity of the total space.
• Defines the extended locus of Hodge classes as the preimage of the zero section under the finite map to $ T(F_{-1}\mathcal{M}) $.

Experimental results

Research questions

• RQ1 How can one compactify the locus of Hodge classes in a family of hyperplane sections when the classical locus has positive dimension or degenerates in the limit?
• RQ2 What is the correct analytic space that captures integral classes becoming Hodge classes in the limit, especially in the presence of singular fibers?
• RQ3 Can the extended locus be constructed in a canonical way using Hodge module theory and remain finite and proper over the base?
• RQ4 What geometric properties does the extended parameter space possess, particularly regarding holomorphic convexity and compact analytic subsets?
• RQ5 How does the ampleness of the cohomological bundle influence the global geometry of the extended locus?

Key findings

• The extended locus of Hodge classes is a finite and proper analytic subspace over the base manifold, providing a canonical compactification of the classical Hodge locus.
• The total space $ \tilde{T}_{\mathbb{Z}}(K) $ is holomorphically convex, a consequence of $ F_{-1}\mathcal{M} $ being a quotient of an ample vector bundle.
• All compact analytic subsets of dimension at least one in $ \tilde{T}_{\mathbb{Z}}(K) $ are contained within the extended locus of Hodge classes.
• The construction realizes the limit of vanishing cycles as Hodge classes in the limit mixed Hodge structure, resolving a key issue in degeneration phenomena.
• The extended parameter space is embedded into the dual of an ample vector bundle, ensuring global generation and geometric control.
• The result confirms a prediction by Clemens regarding the holomorphic convexity of such parameter spaces in the context of hyperplane sections of Calabi-Yau threefolds.