The Cech filtration and monodromy in log crystalline cohomology

Abstract

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