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

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.

💡 Research Summary

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The paper addresses the longstanding p‑adic deformation problem for algebraic K‑theory in the context of semistable schemes, extending the smooth‑case results of Antieau‑Mathew‑Morrow‑Nikolaus (AMMN) to the more singular semistable setting. The authors begin by fixing a complete discretely valued mixed‑characteristic field K with ring of integers OK and perfect residue field k, and they equip OK with its canonical log structure OK♯. A semistable scheme X over OK is locally modeled by algebras of the form OK