Integral Variation of Hodge Structure

Updated 11 December 2025

Integral variation of Hodge structure is a concept defining a lattice with holomorphic filtrations and a polarization, bridging complex-analytic and arithmetic aspects.
It governs period maps into domains like Hermitian symmetric spaces, providing deep insights into deformation theory and the geometry of moduli spaces.
The theory underpins functional transcendence results and arithmetic applications, offering practical tools for studying integrable systems and algebraic families.

An integral variation of Hodge structure (integral VHS) is a highly structured geometric object encoding how the Hodge decomposition of a lattice in the cohomology of a family of complex algebraic or Kähler manifolds varies holomorphically over a base. The integral data, realized as a locally constant sheaf of free abelian groups with additional Hodge-theoretic filtrations and compatible with a polarization, rigidly couples complex-analytic and arithmetic structure. Integral VHS play a central role in Hodge theory, algebraic geometry, arithmetic geometry, and mathematical physics, governing period maps, moduli spaces, arithmetic transcendence, and special families such as Calabi–Yau and abelian varieties.

1. Definition and Basic Properties

An integral VHS of weight $n$ on a complex manifold $B$ consists of:

A local system $\mathcal{H}_\mathbb{Z}$ of free $\mathbb{Z}$ -modules on $B$ (lattice structure), whose complexification $\mathcal{H} = \mathcal{H}_\mathbb{Z} \otimes_\mathbb{Z} \mathcal{O}_B$ carries a flat (Gauss–Manin) connection $\nabla: \mathcal{H} \to \mathcal{H} \otimes \Omega^1_B$ .
A holomorphic, decreasing filtration by subbundles $F^p \mathcal{H}$ (the Hodge filtration), such that for each point $b \in B$ , the pair $(\mathcal{H}_{\mathbb{Z},b}, \{F^p \mathcal{H}_b\})$ is a pure $\mathbb{Z}$ -Hodge structure of weight $n$ .
Griffiths transversality: $\nabla F^p \subset F^{p-1} \otimes \Omega^1_B$ .
A flat bilinear form (polarization) $Q: \mathcal{H}_\mathbb{Z} \otimes \mathcal{H}_\mathbb{Z} \to \mathbb{Z}$ satisfying Hodge–Riemann bilinear relations.

This package constrains both holomorphic (complex-analytic) and integral (arithmetic) aspects, and governs variations coming from geometry—primarily, the cohomology of algebraic varieties in algebraic families (Filippini et al., 2014, Bakker et al., 2017).

2. Period Domains and Period Maps

Fixing the lattice $H_\mathbb{Z}$ , polarization $Q$ , and Hodge numbers $h^{p,q}$ , one considers the period domain $D$ , parameterizing filtrations (Hodge flags) satisfying the Hodge–Riemann relations. For typical cases, $D$ is a Hermitian symmetric domain or a homogeneous open subset of a flag variety, with complex Lie group $G = \mathrm{Aut}(H_\mathbb{R}, Q)$ and $D = G(\mathbb{R})/V$ , $V$ being the stabilizer of a reference flag.

Given an integral VHS over $B$ with monodromy $\Gamma \subset \mathrm{Aut}(H_\mathbb{Z}, Q)$ , the period map $\Phi: B \to \Gamma\backslash D$ sends $b$ to the (class of the) Hodge filtration at $b$ . By Griffiths transversality, the image of $\Phi$ is a horizontal (Griffiths-transverse) submanifold of $D$ . The period map encodes how the Hodge structure deforms in the family, realizing a deep link between geometry and representation theory (Filippini et al., 2014, Bakker et al., 2017).

3. Global Bounds on the Dimension of Period Image

The dimension of the image of the period map for a global integral VHS reflects both local deformation theory and powerful global constraints. For period domains with "level" (the minimal $|p-q|$ with $h^{p,q} \neq 0$ ) at least $3$ and generic $\mathbb{Q}$ -simple Mumford–Tate group, the new global bound is

$\dim \operatorname{Im} \Phi \leq m_{\mathrm{HL}}(G, D)$

where $m_{\mathrm{HL}}(G, D)$ is the minimal codimension drop for any strict Hodge subdatum of $(G, D)$ , minus $1$. In period domains arising from fixed Hodge numbers, $m_{\mathrm{HL}}$ is linear in the $h^{p,q}$ , a notable improvement over the classical (infinitesimal) Carlson–Toledo bound $m_{\mathrm{CT}}$ , which is quadratic in the Hodge numbers.

The theoretical mechanism is that for level $\geq 3$ , global geomorphic and Hodge-theoretic obstructions prevent the period image from sweeping out high-dimensional nilpotent orbits; fixing a single additional rational Hodge vector reduces the dimension in a way governed explicitly by the Hodge numbers. This bound is not just conceptual but attained: the period map associated to the primitive cohomology of universal sextic fourfolds in $\mathbb{P}^5$ has image of dimension exactly $h^{3,1} = 426$ , matching the linear $m_{\mathrm{HL}}$ and greatly outperforming the previous quadratic bounds (Khelifa, 9 Dec 2024).

4. Functional Transcendence and Arithmetic Applications

Integral variations of Hodge structure underpin powerful results on functional transcendence and unlikely intersections. The Ax–Schanuel theorem for integral VHS establishes that any "unexpectedly" large intersection between a period map and an algebraic subvariety must be explained by a reduction in the generic Mumford–Tate group. More technically, if the dimension inequality

$\operatorname{codim}_{S \times \check{D}}(U) < \operatorname{codim}(V) + \operatorname{codim}(W)$

holds, $U$ must project into a weak Mumford–Tate locus. This constrains transcendence degrees and algebraic relations among periods, forming the analytic-transcendence backbone for Zilber–Pink conjectures and arithmetic finiteness theorems for rational points in large VHS (Bakker et al., 2017).

5. Geometric and Physical Realizations

Integral VHS of low weight (especially $\pm 1$ ) naturally encode families of abelian varieties and their duals. For weight 1, the associated family of complex tori arises as the Jacobian fibration

$J(V) \to B, \qquad J(V)_b = V_b / (F^1 V_b + V_{\mathbb{Z},b})$

and a global section yields a symplectic structure whose Lagrangian algebraic fibration realizes a complex integrable system. Notably, the ADE Hitchin system and its generalizations to other Lie types can be canonically reconstructed from weight 1 Z–VHS over the Hitchin base. For BCFG types this construction uses non-compact Calabi–Yau threefolds and GL(2)-orbifold techniques. The equivalence

$\text{(Z–VHS of weight 1) + Seiberg–Witten section} \;\Longleftrightarrow\; \text{Algebraic integrable system}$

provides a remarkable link between Hodge theory, geometric representation theory, and mathematical physics (Langlands duality), illuminating the structure of integrable systems via period geometry (Beck, 2017).

6. Non-Geodesic and Maximal-Dimensional Realizations

There exist variations of Hodge structure of maximal possible dimension whose integral images are not totally geodesic submanifolds in the period domain. For example, the variation associated to the second cohomology of weighted degree 10 hypersurfaces in $\mathbb{P}(1,1,2,5)$ yields a maximal (28-dimensional) horizontal subvariety in the holomorphic contact manifold $SO(4,28)/(U(2)\times SO(28))$ . This integral manifold is nowhere tangent to standard geodesic (Hermitian symmetric) orbits, demonstrating the existence of period images of maximal dimension that are not derived from classical Lie-theoretic constructions. This controls both the local and global differential geometry of VHS and impacts the paper of contact structures in Hodge theory (Carlson et al., 2017).

7. Extension, Degeneration, and the Role of Hodge Modules

Integral VHS extend to singular bases via the theory of Hodge modules. Saito's period-integral map $\mathcal{P}_M$ attached to a polarized Hodge module recovers the classical period pairing (integration of holomorphic forms against integral cycles) and encodes the integral local system data, the Hodge filtration, and period integrals even across singular loci. The minimal extension theorem ensures that these data extend uniquely (in the category of perverse sheaves and filtered $\mathcal{D}$ -modules), and that near degenerations the limiting mixed Hodge structure arises canonically. This structure controls period behavior in singular degenerations, prefiguring phenomena in mirror symmetry, enumerative geometry, and the paper of limiting mixed Hodge structures (Flapan et al., 2019, Sabbah et al., 2022).