Hyodo-Kato theory with syntomic coefficients

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.