Hyodo-Kato theory with syntomic coefficients

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 the study of syntomic coefficients, the rigid analytic Hyodo-Kato map which for the trivial coefficient was studied by Ertl and the author plays a key role. K (ϕ, N )), and there exists a canonical isomorphism). Because of Theorem 0.3 (3), we may regard the syntomic cohomology as an absolute p-adic Hodge cohomology (a p-adic analogue of the Deligne-Beilinson cohomology).Note that we suppose the properness of X in order to consider Hodge filtrations of the de Rham cohomology, but admit a log structure associated to a horizontal divisor. Therefore we may deal with compactifiable cases.One can also show that, if X is smooth and (E , Φ) is induced from a non-logarithmic overconvergent F -isocrystal, then Ψ π,log is independent of π and log, and coincides with the base change map of the non-logarithmic rigid cohomology (Proposition 8.9). Therefore our theory of syntomic coefficients is compatible with the construction in [2] of the syntomic coefficients of Besser's rigid syntomic cohomology for smooth schemes.Our definitions of syntomic coefficients and the Hyodo-Kato map are useful for direct computation. For example, one may extend the methods of p-adic polylogarithms to suitable group schemes with semistable reduction. In the case of good reduction, the p-adic polylogarithms were constructed as classes of the non-logarithmic version of (rigid) syntomic cohomology with syntomic coefficients, and described explicitly for some special cases [1,3,4,40]. In the case of semistable reduction, the p-adic polylogarithms should be defined as classes of the logarithmic (rigid) syntomic cohomology with syntomic coefficients which we define in §10 of this article. Since our construction is simple and explicit, one may calculate them as Čech-de Rham coycles on p-adic analytic spaces.

Poincaré duality for rigid analytic Hyodo--Kato theory

“…In this section we recall the definition of the log overconvergent site and log overconvergent isocrystals from [31].…”

“…where the product is taken in OC (Y /T ) (see [31,Prop. 2.20] for products in the log overconvergent site).…”

“…Therefore, it is suitable not only for explicit computation but also for generalisations. For example, the second author studied coefficients for rigid analytic Hyodo-Kato theory in [31], and generalised the additional structures on the rigid Hyodo-Kato cohomology and the Hyodo-Kato map to cohomology with coefficients. The construction of rigid Hyodo-Kato theory with compact support in the present article can be seen as an application of [31].…”

“…For example, the second author studied coefficients for rigid analytic Hyodo-Kato theory in [31], and generalised the additional structures on the rigid Hyodo-Kato cohomology and the Hyodo-Kato map to cohomology with coefficients. The construction of rigid Hyodo-Kato theory with compact support in the present article can be seen as an application of [31]. More precisely, we introduce several types of log rigid cohomology with compact support as log rigid cohomology with coefficients in a certain type of log overconvergent isocrystals.…”

“…Outline of this paper. In §1, we recall notions concerning log overconvergent (F -)isocrystals which were introduced in [31]. In particular, we recall definitions of the log rigid cohomology and the rigid Hyodo-Kato cohomology with coefficients in log overconvergent (F -)isocrystals.…”