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On the $p$ -adic deformation problem for the $K$ -theory of semistable schemes

This paper establishes a semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square to solve the $p$ -adic deformation problem for continuous K-theory by linking algebraic K-theory of semistable schemes to logarithmic topological cyclic homology via the Hyodo-Kato Chern character, thereby providing a purely K-theoretic proof of Yamashita's semistable $p$ -adic Lefschetz $(1,1)$ -theorem.

Original authors: Federico Binda, Tommy Lundemo, Alberto Merici, Doosung Park

Published 2026-05-25

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CC BY 4.0

Original authors: Federico Binda, Tommy Lundemo, Alberto Merici, Doosung Park

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to restore a historic building that has been damaged by time. You have a perfect blueprint of the building as it looked in its prime (the "generic fiber"), and you have the crumbling ruins of the foundation (the "special fiber"). Your goal is to figure out if a specific design element found in the ruins can be successfully rebuilt to match the original blueprint.

This paper, written by Binda, Lundemo, Merici, and Park, tackles a very similar problem, but in the abstract world of algebraic geometry and number theory. They are trying to solve a "deformation problem": Can we lift a mathematical object from a "bad" or "damaged" state back up to a "perfect" or "smooth" state?

Here is a breakdown of their work using simple analogies:

1. The Setting: The "Bad" vs. "Good" Building

In mathematics, fields of numbers can be "smooth" (like a polished marble floor) or "semistable" (like a floor with cracks that follow a predictable pattern, like a grid).

The Smooth Case: Previous mathematicians (Antieau, Mathew, Morrow, and Nikolaus) had already figured out how to lift designs from a smooth floor to a perfect one. They used a "magic square" (a mathematical diagram) that acted like a bridge. If a design element fit in the right spot on the bridge, it could be lifted.
The Semistable Case (The Problem): The authors ask: What if the floor is cracked in a specific, "semistable" way? The old bridge doesn't work anymore because the cracks change the rules of the game. They needed to build a new bridge specifically for these cracked, semistable floors.

2. The New Bridge: The "Logarithmic" Fiber Square

To build this new bridge, the authors use a tool called Log Geometry.

The Analogy: Imagine standard geometry is like looking at a building with just your eyes. Log geometry is like putting on special glasses that let you see the "logarithmic structure"—essentially, the hidden "skeleton" or the "growth rings" of the cracks.
The Innovation: The authors created a new version of the "magic square" (called a fiber square) that works with these special glasses. This new square connects the K-theory (the "design elements") of the cracked floor to a new type of mathematical measurement called Logarithmic Topological Cyclic Homology.
The Result: They proved that this new square is solid. It successfully relates the "bad" state to the "good" state, but only if you use the right tools (the log structure).

3. The Key to the Lock: The Hyodo-Kato Chern Character

Once they built the bridge, they needed a way to check if a specific design element could actually cross it.

The Obstruction: In the smooth case, there was a known "test" (the Hodge filtration) to see if a design would fit. In the semistable case, the test was vague and hard to understand.
The Solution: The authors discovered that the "test" is actually governed by something called the Hyodo-Kato Chern character.
The Metaphor: Think of the Hyodo-Kato Chern character as a specialized scanner. When you scan a design element from the cracked floor, this scanner tells you exactly where it belongs in the perfect blueprint.
- If the scanner says the element fits into the "filtered" section of the blueprint, success! You can lift it.
- If it doesn't fit, the element cannot be lifted.
The Claim: The paper proves that this scanner is the only thing that matters. If the Hyodo-Kato Chern character passes the test, the lift is possible. This solves a long-standing question about how to lift these mathematical objects in the semistable case.

4. The Application: The "Lefschetz (1,1)-Theorem"

The authors didn't just build the bridge; they drove a car across it to prove it works.

The Achievement: They used their new bridge and scanner to provide a purely K-theoretic proof of a famous theorem by Yamashita (the semistable p-adic Lefschetz (1,1)-theorem).
Why it matters: Previously, this theorem was proven using different, more complicated methods. The authors showed that their new "K-theory bridge" could reach the same destination, confirming that their new mathematical framework is powerful and correct.

5. A Side Trip: The "Characteristic Zero" Story

In the final section, the authors briefly look at a different world where the numbers behave differently (characteristic zero, like working with standard real numbers instead of modular arithmetic).

They showed that their method of using "logarithmic glasses" to fix cracked structures also works in this different world, generalizing a result by another mathematician named Morrow. This confirms that their approach is robust and versatile, not just a one-trick pony for the p-adic world.

Summary

In short, this paper is about repairing a broken mathematical bridge.

The Problem: The old bridge (for smooth shapes) didn't work for cracked, semistable shapes.
The Fix: The authors built a new bridge using "logarithmic glasses" (log geometry) and a new type of measurement (logarithmic cyclic homology).
The Test: They found that a specific scanner (the Hyodo-Kato Chern character) determines exactly which parts of the cracked shape can be restored to the perfect shape.
The Proof: They used this new system to solve a famous puzzle (Yamashita's theorem), proving their new bridge is strong enough to carry the weight of important mathematical truths.

Technical Summary: On the p-adic Deformation Problem for the K-Theory of Semistable Schemes

1. The Problem
The paper addresses the $p$ -adic deformation problem for algebraic K-theory in the context of semistable reduction. Specifically, it seeks to characterize the obstruction to lifting a K-theory class $x \in K_0(X_k; \mathbb{Q})$ from the special fiber $X_k$ of a scheme $X$ to the continuous K-theory $K_{cts}^0(X; \mathbb{Q})$ of the generic fiber.

In the case of good reduction (smooth and proper schemes over a complete discretely valued field $K$ ), Antieau, Mathew, Morrow, and Nikolaus [1] established that this lifting is governed by the Hodge filtration on the de Rham cohomology of the generic fiber, via a cartesian fiber square relating continuous K-theory to topological cyclic homology (TC) and crystalline invariants. However, in the semistable case (bad reduction), the standard crystalline Chern character and the associated obstruction classes lack a direct crystalline interpretation, and the relationship between K-theory and logarithmic invariants was previously unclear. The authors aim to extend the smooth results to the semistable setting using logarithmic geometry.

2. Methodology
The authors employ a synthesis of logarithmic geometry, motivic homotopy theory, and trace methods:

Logarithmic Generalization of Fiber Squares: The authors generalize the cartesian square of Antieau–Mathew–Morrow–Nikolaus to the logarithmic setting. They utilize Rognes' logarithmic topological Hochschild homology ( $\text{log-THH}$ ) and the logarithmic Hochschild–Kostant–Rosenberg (HKR) filtration.
Vertical Log Structures: A crucial technical step involves analyzing "vertical" log structures (where the cokernel of the monoid map is a group). The authors prove that for vertical, regular, and log regular algebras, the rationalized logarithmic cyclic homology invariants behave similarly to the trivial log structure case, allowing them to "merge" the smooth and logarithmic fiber squares.
Hyodo–Kato Chern Character: The authors construct a Hyodo–Kato Chern character ( $ch_{HK}$ ) from the K-theory of the special fiber to the Hyodo–Kato cohomology. This is achieved using Chern orientations in logarithmic motivic homotopy theory and applying them to the logarithmic de Rham–Witt complex.
Trace Maps and Homotopy K-Theory: To handle the interaction between K-theory and log structures, the authors replace standard K-theory with homotopy K-theory ($KH$) where necessary, utilizing the log cyclotomic trace. They establish a commutative diagram relating $K$ , $KH$, and logarithmic TC.
Splitting Filtrations: Using the Hyodo–Kato isomorphism, which relates the de Rham cohomology of the generic fiber to the Hyodo–Kato cohomology of the special fiber, the authors split the rationalized filtrations of negative cyclic homology ( $HC^-$ ) and periodic cyclic homology ($HP$).

3. Key Contributions and Results

The Logarithmic Bloch–Esnault–Kerz Square (Theorem A): The authors establish a cartesian square of spectra for a vertical, regular, log regular $O_K^\sharp$ -algebra $(R, P)$ :
$\begin{array}{ccc} K(R; \mathbb{Q}_p) & \to & K(R/p; \mathbb{Q}_p) \\ \downarrow & & \downarrow \\ HC^-((R, P)/O_K^\sharp; \mathbb{Q}_p) & \to & HP((R, P)/O_K^\sharp; \mathbb{Q}_p) \end{array}$
This relates ordinary K-theory to logarithmic cyclic homology invariants.
The Hyodo–Kato Deformation Criterion (Theorem B): For a proper flat scheme $X$ over $O_K$ with semistable special fiber $X_k$ , a class $x \in K_0(X_k; \mathbb{Q})$ lifts to $K_{cts}^0(X; \mathbb{Q})$ if and only if its image under the Hyodo–Kato Chern character, $ch_{HK}(x)$ , lies in the Hodge filtration $\bigoplus_{i=0}^\infty \text{Fil}^{\ge i} H^{2i}_{dR}(X_K; \mathbb{Q}_p)$ under the Hyodo–Kato equivalence. This identifies the obstruction to lifting explicitly with the Hyodo–Kato Chern character.
Logarithmic K-Theory Variant (Theorem C): The authors provide a variant of the fiber square involving homotopy K-theory ($KH$) of the special fiber and logarithmic TC, bridging the gap between algebraic K-theory and logarithmic trace invariants.
Recovery of Yamashita's Theorem (Section 6): As a direct application, the authors provide a purely K-theoretic proof of Yamashita's semistable $p$ -adic Lefschetz $(1,1)$ -theorem. They show that a line bundle class in $\text{Pic}(X_k; \mathbb{Q})$ lifts to $\text{Pic}(X; \mathbb{Q})$ if and only if its first Hyodo–Kato Chern class lies in $\text{Fil}^{\ge 1} H^2_{dR}(X_K; \mathbb{Q}_p)$ .
Characteristic Zero Generalization (Section 7): The authors extend these methods to the characteristic zero setting (schemes over $K[[t]]$ ), proving a semistable generalization of a theorem by Morrow regarding the liftability of K-theory classes in terms of the flat filtration on relative logarithmic de Rham cohomology.

4. Significance
The paper claims to provide the first K-theoretic application of logarithmic trace invariants of genuinely geometric origin. By establishing the Hyodo–Kato Chern character as the governing obstruction for the $p$ -adic deformation problem in the semistable case, the work extends the foundational results of Antieau–Mathew–Morrow–Nikolaus from the smooth to the semistable setting.

The authors note that their results offer a more flexible and transparent relationship between algebraic K-theory and logarithmic trace invariants, potentially advancing the conjectures regarding the relationship between K-theory and logarithmic cyclic homology (specifically [30, Conjecture 1.5]). The recovery of Yamashita's theorem via these methods demonstrates the utility of the constructed fiber squares in classical arithmetic geometry problems. The work is presented as a significant step toward a unified understanding of K-theory in the presence of semistable reduction, though the authors remain modest regarding the full scope of future applications, noting that further progress is needed to fully resolve the relationship between K-theory and logarithmic invariants.