Semistable p-adic Lefschetz (1,1) Theorem

• The theorem characterizes when a line bundle or divisor on the special fiber lifts to the total space by requiring its Hyodo–Kato class to lie in the F¹ step of the Hodge filtration.
• It integrates log-crystalline, de Rham–Witt, and K-theoretic methods to bridge log-motivic cohomology with arithmetic geometry in semistable settings.
• Proof techniques employ exact sequences, deformation theory, and log de Rham–Witt calculations to address obstructions in lifting cycle classes under semistable reduction.

The semistable $p$-adic Lefschetz $(1,1)$-theorem is a generalization of the classical Lefschetz $(1,1)$-theorem, formulated in the context of semistable degenerations over $p$-adic bases. It characterizes the obstruction to lifting line bundles (and more generally, divisors and cohomology classes) from the special fiber of a semistable scheme to its total space, in terms of Hodge-theoretic and cohomological filtrations. Central to its formulation and proof are logarithmic structures, log-motivic cohomology, log de Rham–Witt theory, crystalline and rigid cohomology, and, in modern developments, $K$-theory and topological cyclic homology. The result can be seen as a $p$-adic analog, in the semistable case, of the variational Hodge conjecture’s (1,1)-part.

1. Logarithmic and Cohomological Framework

Let $X$ be a proper flat scheme over $W(k)$ or over a complete discrete valuation ring $O_K$ of mixed characteristic $(0,p)$, with reduced special fiber $Y$ (“semistable reduction”) and generic fiber $X_K$. The logarithmic structure is induced by the special fiber, resulting in fine saturated log schemes and log-smooth morphisms. The log-motivic cohomology group in bidegree $(2,1)$ is defined as

$$H^{2,1}_{\mathrm{log-mot}}(X) := H^2(X, \mathbb{Z}_{\log}(1)),$$

where $\mathbb{Z}_{\log}(1)$ is a complex of Zariski sheaves whose only nontrivial cohomology sheaf is the modified log-structure $N^p$ on $X$; this group is canonically isomorphic to $H^1(X, N^p)$ and to the logarithmic Milnor $K$-group in weight 1 on each open $U$ in $X$ [(Gregory et al., 2021), §2–3].

For $n=1$, $H^{2,1}_{\mathrm{log-mot}}(X)$ coincides with the logarithmic Picard group, the group of isomorphism classes of line bundles on the log scheme $(X, M_X)$ [(Gregory et al., 2021), Prop. 2.13]. The Hyodo–Kato cohomology $H^2_{\mathrm{HK}}(X/W(k))_{\mathbb{Q}}$—described by the log-crystalline cohomology of $(X, M_X)$—admits a natural Frobenius and monodromy operator, and is equipped with a Hodge filtration; via the Hyodo–Kato isomorphism

$$H^2_{\mathrm{dR}}(X_K/K) \cong H^2_{\mathrm{HK}}(X/W(k))_\mathbb{Q} \otimes_{K_0} K,$$

the Hodge filtration on de Rham cohomology descends to a canonical filtration on $H^2_{\mathrm{HK}}$ [(Gregory et al., 2021), §1b; (Binda et al., 20 Jan 2026)].

A key role is played by the Hyodo–Kato (log-crystalline) Chern class map

$$c_1^{\mathrm{HK}} : \mathrm{Pic}(Y_k) \to H^2(Y_k, W \Omega^1_{Y_k/(k, \mathbb{N})}) \otimes_{W(k)[1/p]} K,$$

and its compatibility with de Rham classes via the Hyodo–Kato isomorphism.

2. Statement of the Semistable $p$-adic Lefschetz $(1,1)$ Theorem

Let $X$ be a proper semistable scheme as above, $Y$ its special fiber, and let $$\alpha \in H^{2,1}_{\mathrm{log-mot}}(X)_{\mathbb{Q}}$$ (or, for the line bundle case, $x \in \mathrm{Pic}(Y_k) \otimes \mathbb{Q}$). The semistable $p$-adic Lefschetz $(1,1)$ theorem asserts:

Theorem:

$\alpha$ lifts to a class in the continuous pro-cohomology $$H^2_{\mathrm{cont}}(X_\bullet, \mathbb{Z}_{\log}(1))_{\mathbb{Q}}$$ if and only if its Hyodo–Kato class $cl_{\mathrm{HK}}(\alpha)$ lies in the $F^1$-step of the Hodge filtration,

$$cl_{\mathrm{HK}}(\alpha) \in \mathrm{Fil}^1 H^2_{\mathrm{HK}}(X/W(k))_{\mathbb{Q}}.$$

For the Picard group,

$$x \in \mathrm{Pic}(Y_k) \otimes \mathbb{Q} \text{ lifts to } \mathrm{Pic}(Y) \otimes \mathbb{Q} \iff \rho_\pi(c_1^{\mathrm{HK}}(x)) \in \mathrm{Fil}^1 H^2_{\mathrm{dR}}(Y_K/K).$$

[(Gregory et al., 2021), Thm.; (Binda et al., 20 Jan 2026), Thm. \ref{thm:Lefschetz11}; (Lazda et al., 2017), Theorem (semistable Lefschetz (1,1))].

3. Cohomological, $K$-theoretic, and Logarithmic Perspectives

Multiple frameworks unify in the semistable $p$-adic Lefschetz $(1,1)$ theorem:

• Log Crystalline and de Rham–Witt theory: The construction of the log de Rham–Witt sheaves $W_r \omega^1_{Y_k^\times/k^\times, \log}$ and associated exact sequences replaces de Rham complexes and encodes contributions from singularities on the special fiber [(Lazda et al., 2017), §2–3; (Gregory et al., 2021), §1c].
• $K$-theoretic approach: Binda–Lundemo–Merici–Park provide a purely $K$-theoretic proof by constructing a “logarithmic” Beilinson–Bloch–Esnault–Kerz (BBEK) fiber square, relating continuous $K$-theory of $Y_K$ with the homotopy $K$-theory of $Y_k$ and log-cyclotomic trace, and matching obstructions to lifting with the Hyodo–Kato Chern character [(Binda et al., 20 Jan 2026), Thm. \ref{thm:Lefschetz11}, \ref{thm:hkchern2}]. The Hyodo–Kato Chern class serves as a trace-theoretic obstruction precisely corresponding to the Hodge filtration on the relevant de Rham cohomology group.
• Cycle-theoretic and rigid cohomology methods: The rigid cohomology analog for varieties over equicharacteristic $p$-adic bases expresses the condition as the inclusion of the first Chern class in the overconvergent $(\varphi,\nabla)$-lattice in the Robba ring. Thus, the obstruction to lifting a line bundle is the requirement that its class lies in $H^2_{\mathrm{rig}}(X/\mathcal{E}^\dagger)$, reflecting a crystalline-rigid “Hodge” condition [(Lazda et al., 2017), Theorem].

The equivalence of $K$-theoretic, log-motivic, and crystalline perspectives is supported by various comparison theorems and commutative diagrams, e.g., the Picard—first Chern class commutative square [(Binda et al., 20 Jan 2026), Proposition \ref{Pic.4}].

4. Techniques and Proof Strategies

The proof proceeds in several stages:

• Exact sequences and deformation theory: The obstruction to lifting is governed by a long exact sequence derived from gluing $\mathbb{Z}_{\log}(1)$ on $Y$ with log-syntomic complexes. The map between $H^2(Y, \mathbb{Z}_{\log}(1))$ and $H^3_{\mathrm{cont}}(\mathrm{s.Og} X_\bullet(1))$ captures the unique obstruction; this map is identified with the projection to the $F^2$-step of the Hodge filtration, so the obstruction vanishes precisely when the Hyodo–Kato class is contained in $\mathrm{Fil}^1$ [(Gregory et al., 2021), §3].
• $K$-theoretic filtration splitting: The log–BBEK fiber square relates $K$-theory spectra and cyclic homology explicitly, and passage to $\pi_0$ computes the obstructions in terms of the Hodge filtration on de Rham cohomology [(Binda et al., 20 Jan 2026), §3]. Log-motivic homotopy theory bridges the obstruction from $K$-theory to Hyodo–Kato theory.
• Elementary log de Rham–Witt calculations: In equicharacteristic $p$, diagram chases and exactness properties of log de Rham–Witt complexes yield injectivity and surjectivity needed for reduction to the Hodge condition, and facilitate extension to global fields and function fields [(Lazda et al., 2017), Step 1–4].
• Specialization to the Picard group: For $n=1$, the criterion specializes to the liftability of logarithmic line bundles, with the Hyodo–Kato first Chern class governing the obstruction [(Gregory et al., 2021), §4; (Binda et al., 20 Jan 2026), §4; (Lazda et al., 2017), Theorem].

5. Corollaries, Examples, and Limitations

• Corollaries: The $K$-theoretic approach extends the theorem to all higher $K$-groups for semistable families and generalizes the deformational Hodge conjecture to characteristic 0 and equicharacteristic $p$ [(Binda et al., 20 Jan 2026), §5].
• Global results: Algebraicity lifting for cohomology classes on varieties over global function fields is achieved by relating the local lifting criterion at places of semistable reduction with the global Picard group structure [(Lazda et al., 2017), §5a].
• Counterexamples: The theorem fails when $\mathbb{Q}_p$-coefficients replace $\mathbb{Q}$ in the Picard group. Concrete examples with supersingular elliptic curves display that not every crystalline class in the overconvergent lattice arises as a Chern class of an actual line bundle [(Lazda et al., 2017), §5b].
• Analytic examples: For standard semistable schemes defined by equations like $X_1 \cdots X_r = \pi$, $K$-theoretic methods provide new “analytic” Lefschetz theorems for hypersurfaces with degenerations [(Binda et al., 20 Jan 2026), §5].

6. Comparison with Good Reduction and Classical Theory

In the case of good reduction, the obstruction criterion reduces to checking the first Chern class lies in the Hodge filtration $F^1H^2_{\mathrm{dR}}$. The semistable case requires replacing the usual de Rham complexes with their logarithmic analogs, and monodromy plays a discernible role. In mixed characteristic, the Berthelot–Ogus criterion involves the classical Hodge filtration, whereas, in the semistable or logarithmic case, the “Hodge condition” is interpreted in log-crystalline, overconvergent, or $K$-theoretic terms depending on the context (Gregory et al., 2021, Binda et al., 20 Jan 2026, Lazda et al., 2017).

Logarithmic phenomena arising in the semistable setting include new exact sequences for log de Rham–Witt sheaves, the monodromy-killed subspaces, and new obstructions connected to boundary maps in long exact deformation sequences—distinct from the smooth case.

7. Impact and Further Developments

The semistable $p$-adic Lefschetz $(1,1)$ theorem unifies approaches from $K$-theory, motivic cohomology, log-crystalline and rigid cohomology, and provides a template for further research in $p$-adic Hodge theory, degeneration, and period comparison problems. Its proof methods have influenced subsequent results in $K$-theoretic Lefschetz theorems for higher $K$-groups and the study of $p$-adic cycles, and have clarified the role of filtrations, monodromy, and log structures in arithmetic geometry (Gregory et al., 2021, Binda et al., 20 Jan 2026, Lazda et al., 2017).

A plausible implication is that these frameworks will underpin advances in $p$-adic period maps and allow sharper control over cycle class liftability and the structure of $p$-adic cycle classes in semistable and singular geometry.