Hyodo-Kato theory with syntomic coefficients
Kazuki Yamada
TL;DR
The paper constructs a comprehensive p-adic framework for cohomology with coefficients in variations of mixed Hodge structures, blending Hyodo–Kato theory with log overconvergent $F$-isocrystals and syntomic coefficients. It develops Hyodo–Kato cohomology with coefficients, proves rigidity results via a tannakian approach to unipotent objects, and extends these constructions to mixed characteristic through $p$-adic Hodge and syntomic cohomologies, including a generalized Hyodo–Kato map. A Kim–Hain CDGA with monodromy and Frobenius structures underpins the coefficient theory, while base-change and independence results guarantee robustness under changes of log branches and uniformizers. Collectively, these results provide a solid foundation for explicit p-adic realizations of Hodge-type theories and for computations of p-adic regulators and special values in the semistable setting.
Abstract
The purpose of this article is to establish theories concerning $p$-adic analogues of Hodge cohomology and Deligne-Beilinson cohomology with coefficients in variations of mixed Hodge structures. We first study log overconvergent $F$-isocrystals as coefficients of Hyodo-Kato cohomology. In particular, we prove a rigidity property of Hain-Zucker type for mixed log overconvergent $F$-isocrystals. In the latter half of the article, we give a new definition of syntomic coefficients as coefficients of $p$-adic Hodge cohomology and syntomic cohomology, and prove some fundamental properties concerning base change and admissibility. In particular, we see that our framework of syntomic coefficients depends only on the choice of a branch of the $p$-adic logarithm, but not on the choice of a uniformizer of the base ring. The rigid analytic reconstruction of Hyodo-Kato map studied by Ertl and the author plays a key role throughout this article.