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

TL;DR

This paper extends the understanding of the $p$-adic deformation of algebraic K-theory for semistable schemes by relating it to logarithmic topological cyclic homology and solving the deformation problem.

Contribution

It establishes a semistable fiber square relating K-theory to logarithmic topological cyclic homology and solves the $p$-adic deformation problem in this context.

Findings

Obstruction to lifting K-theory classes is governed by the Hyodo-Kato Chern character.

Provides a K-theoretic proof of Yamashita's semistable $p$-adic Lefschetz $(1,1)$-theorem.

Extends deformation results to semistable schemes, generalizing prior work.

Abstract

We establish a semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square, relating the algebraic K-theory of a semistable scheme to its logarithmic topological cyclic homology. We prove that the obstruction to lifting K-theory classes is governed by the Hyodo-Kato Chern character. This answers the $p$-adic deformation problem for continuous K-theory in the semistable case, extending the work of Antieau-Mathew-Morrow-Nikolaus. As an application, we provide a purely K-theoretic proof of Yamashita's semistable $p$-adic Lefschetz $(1,1)$-theorem.