Hyodo-Kato理论及其合成系数
Hyodo-Kato theory with syntomic coefficients
AI总结：
本文建立混合Hodge结构系数下的$p$-进Hodge与合成上同调理论，定义合成系数并证明其基变换性质，依赖$p$-进对数分支而非均匀化子。
摘要：
本文旨在建立关于混合Hodge结构变形的系数下的Hodge上同调和Deligne-Beilinson上同调的$p$-进类比理论。我们首先研究对数过收敛$F$-等晶体作为Hyodo-Kato上同调的系数。特别地，我们证明了混合对数过收敛$F$-等晶体的Hain-Zucker型刚性性质。在文章的后半部分，我们给出了合成系数的新定义，作为$p$-进Hodge上同调和合成上同调的系数，并证明了关于基变换和可容许性的一些基本性质。特别是，我们看到我们的合成系数框架仅依赖于$p$-进对数的一个分支的选择，而不依赖于基环的均匀化子的选择。Ertl和作者研究的Hyodo-Kato映射的刚性解析重构在本文中起着关键作用。
英文摘要：
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.
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