Liu–Zhu Period Map in p-adic Hodge Theory

• Liu–Zhu period map is a refined p-adic morphism that links local systems with nilpotent Higgs bundles through specialized period sheaves.
• It employs the pro-étale topology and Faltings’ extension to create a functorial link between étale and de Rham cohomology data.
• The construction unifies local computations and differential invariants, underpinning modern p-adic nonabelian Hodge theory.

The Liu–Zhu period map is a refined period morphism in $p$-adic Hodge theory that interrelates the geometry of $p$-adic analytic varieties with arithmetic data via period sheaves and the correspondence between $p$-adic local systems and Higgs bundles. It plays a foundational role in the modern approach to $p$-adic nonabelian Hodge theory, particularly in the context of rigid analytic spaces over nonarchimedean fields and their moduli.

1. Origins and General Framework

The Liu–Zhu period map is developed in the context of $p$-adic Hodge theory for rigid analytic spaces, with a focus on the interplay among analytic, étale, and de Rham data. In [LZ17], the construction relates a $\mathbb{Q}_p$-local system (i.e., a $p$-adic étale local system) on the rigid generic fiber of a formal scheme to a nilpotent Higgs bundle via a period map. The central mechanism involves a period sheaf $\mathcal{O}_\mathcal{C}$ (or its integral variant $\mathcal{O}_\mathcal{C}^\dagger$ as in (Wang, 2021)), whose structure encodes the interpolation between étale and de Rham cohomology.

Key features of the framework include:

• The use of the pro-étale site $X_{\mathrm{pro\acute{e}t}}$ of a rigid variety $X$ over a $p$-adic field.
• The assignment of a functor from the category of $\mathbb{Q}_p$-local systems to the category of nilpotent Higgs bundles via tensoring with the period sheaf and pushforward by the pro-étale to étale topology morphism $\nu_*$.
• The period map itself is a natural transformation attaching to $L$, a $\mathbb{Q}_p$-local system, the Higgs bundle $H(L):=\nu_*(L \otimes \mathcal{O}_\mathcal{C})$.

2. Construction of the Period Sheaf and Higgs Field

The construction in (Wang, 2021) refines the period sheaf to an integrally defined, overconvergent object $\mathcal{O}_\mathcal{C}^\dagger$. The sheaf is defined by inverse limits:

$$\mathcal{O}_\mathcal{C}^\dagger = \varprojlim_\rho \varprojlim_n \mathrm{Sym}^n(E_{(\rho)})$$

where $E_{(\rho)}$ is an integral Faltings' extension built using the $p$-adic cotangent complex after Beilinson and Bhatt. Locally on a small affine chart $X = \mathrm{Spf}(R)$, this extension takes the form:

$$0 \to R^+\{1\} \to E_{(\rho)} \to R^+ \otimes_R \Omega_R^1 \to 0$$

The associated Higgs field $\Theta$ is given locally by

$$\Theta = \sum_{i=1}^d \partial Y_i \otimes d\log T_i$$

with values in $$\mathcal{O}_\mathcal{C}^\dagger \otimes_{\mathcal{O}_X} \Omega_X^1(-1)$$.

3. The Liu–Zhu Period Map as a Functorial Transformation

The Liu–Zhu period map assigns to any $\mathbb{Q}_p$-local system $L$ the Higgs bundle:

$$H(L) := \nu_* (L \otimes \mathcal{O}_\mathcal{C}^{(\dagger)})$$

The Higgs field on $H(L)$ is induced by the natural action on the period sheaf. Remarkably, after inverting $p$, the intricate integral structure becomes compatible with previous constructions, unifying approaches (as in [Faltings], [Hyodo], and later [Abbes–Gros–Tsuji]). The resulting functorial correspondence between local systems and Higgs bundles is a $p$-adic analog of the classical nonabelian Hodge correspondence.

Local computations (with Koszul complexes and group cohomology) establish the quasi-inverse nature of the correspondence, confirming that small $\mathbb{Q}_p$-local systems correspond bijectively to nilpotent Higgs bundles with vanishing $p$-curvature in this context.

4. Differential Properties: Kodaira–Spencer and Sen Morphisms

The Liu–Zhu period map is tightly linked with geometric differential structures. In the inscribed $v$-sheaves framework (Howe, 15 Aug 2025), the period map’s derivative is shown to coincide with the canonical Kodaira–Spencer map. Under local trivialization and splitting of the filtration, one can compare connections $\nabla$ and a constant split connection $\nabla'$; their difference precisely measures the infinitesimal variation encoded by the derivative:

$$d(\pi_\mathrm{Hdg}) = \kappa_{\omega_\nabla^\mathrm{Fil}}$$

Here, for any tangent vector $t$ and representation $V$, one computes $f_{t,V} = \nabla_{t,V} - \nabla'_{t,V}$, which by functoriality glues to a Lie algebra-valued morphism $t \mapsto f_t \in \omega(\mathfrak{g})$, then projects mod $\mathrm{Fil}^0$ to yield the Kodaira–Spencer map. Calculations show that the derivative of the lattice period map in the trivialization agrees with this difference connection. A plausible implication is that further differential invariants of the period map (such as the geometric Sen operator or canonical Higgs field) may be recovered functorially in the inscribed setting, uniquely characterizing the tangent directions in moduli spaces.

5. Connections with Moduli Spaces and Algebraicity

The image and properties of the Liu–Zhu period map are reflected in broader studies of period morphisms (Liu et al., 2016, Song, 2021, Debarre et al., 2017). In such contexts, period maps for moduli spaces of varieties (e.g., polarized hyperkähler manifolds) map onto period domains, with explicit characterization of the complement as unions of Heegner divisors defined by lattice-theoretic conditions. The Liu–Zhu period map shares with these classical period maps the phenomenon that its image is generally not surjective; rather, it avoids loci determined by degenerations, wall divisors, or limiting configurations, which can be described arithmetically or geometrically. Algebraicity results established for period images, such as boundedness, properness, and finite étale covering properties, may transfer to the Liu–Zhu setting, granting meaningful geometric and topological structure to the period image.

6. Extensions, Generalizations, and Further Directions

The Liu–Zhu period map can be situated within a lattice of frameworks:

• In derived algebraic geometry (Natale, 2014), period maps are interpreted as morphisms of derived deformation functors, lifting the classical Griffiths map to a homotopical setting controllable via $L_\infty$ structures.
• In the context of v-sheaves, diamonds, and inscribed structures (Howe, 15 Aug 2025), the period map and its refinements become accessible as smooth maps of spaces with well-behaved tangent bundles and Banach-Colmez geometric structures not visible in previous pure $p$-adic analytic treatments.
• The links with moduli of local shtukas, infinite-level Shimura varieties, and twistor spaces on the Fargues–Fontaine curve suggest that ongoing work will further clarify the connection between $p$-adic periods, geometric Sen theory, and functorial tensor structures.

A plausible implication is that the Liu–Zhu period map provides a universal moduli-theoretic tool for interpolating between arithmetic and geometric invariants of $p$-adic analytic spaces, capturing both local system data and Higgs bundle geometry, and governing their infinitesimal deformations via canonical differential operators (Kodaira–Spencer/Higgs/Sen morphisms).

7. Summary Table: Comparison Across Period Maps

Period Map Type	Target Space	Key Structural Feature
Classical (Griffiths)	Flag variety / period domain	Hodge filtration, variation
Crystalline	$B_\mathrm{crys}$-modules	Comparison via period sheaves
Hyodo–Kato	$B_\mathrm{crys}$-modules	Derived de Rham / Frobenius
Liu–Zhu (p-adic)	Higgs bundles / twistor spaces	Nilpotent Higgs field, Sen map
Inscribed / Twistor	Banach-Colmez spaces, diamonds	Smooth tangent bundles, differential invariants

The Liu–Zhu period map thus encompasses and refines several aspects of period morphisms in arithmetic geometry, leading to deeper understanding of $p$-adic nonabelian Hodge theory, moduli of rigid analytic spaces, and the geometric structures underlying Galois representations and Higgs bundles.