The Cech filtration and monodromy in log crystalline cohomology

Abstract

For a strictly semistable log scheme $Y$ over a perfect field $k$ of characteristic $p$ we investigate the canonical Čech spectral sequence $(C{)}_T$ abutting to the Hyodo-Kato (log crystalline) cohomology $H_{crys}^{\ast }(Y/T{)}_{\mathbb{Q}}$ of $Y$ and beginning with the log convergent cohomology of its various component intersections $Y^i$. We compare the filtration on $H_{crys}^{\ast }(Y/T{)}_{\mathbb{Q}}$ arising from $(C{)}_T$ with the monodromy operator $N$ on $H_{crys}^{\ast }(Y/T{)}_{\mathbb{Q}}$. 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 $H_{dR}^{\ast }(X_K)=H_{dR}^{\ast }(X_K^{an})$.