The Cech filtration and monodromy in log crystalline cohomology

TL;DR

This paper explores the relationship between the Cech filtration, monodromy operator, and log crystalline cohomology of semistable schemes over a perfect field, revealing how geometric structures influence cohomological invariants.

Contribution

It introduces a comparison between the Cech spectral sequence filtration and the monodromy operator in log crystalline cohomology, and relates these to residue maps and analytic structures.

Findings

Comparison of filtrations and monodromy operator in cohomology.

Expression of monodromy via residue maps.

Reflection of simplicial structure in de Rham cohomology.

Abstract

For a strictly semistable log scheme $Y$ over a perfect field $k$ of characteristic $p$ we investigate the canonical \v{C}ech spectral sequence $(C)_T$ abutting to the Hyodo-Kato (log crystalline) cohomology $H_{crys}^*(Y/T)_{\mathbb{Q}}$ of $Y$ and beginning with the log convergent cohomology of its various component intersections $Y^i$. We compare the filtration on $H_{crys}^*(Y/T)_{\mathbb{Q}}$ arising from $(C)_T$ with the monodromy operator $N$ on $H_{crys}^*(Y/T)_{\mathbb{Q}}$. We also express $N$ through residue maps and study relations with singular cohomology. If $Y$ lifts to a proper strictly semistable (formal) scheme $X$ over a finite totally ramified extension of $W(k)$, with generic fibre $X_K$, we obtain results on how the simplicial structure of $X_K^{an}$ (as analytic space) is reflected in $H_{dR}^*(X_K)=H_{dR}^*(X_K^{an})$.