Some examples of DR-indecomposable special fibers of semi-stable reductions over Witt rings

TL;DR

This paper demonstrates that certain semi-stable reductions over Witt rings are not DR-decomposable, providing counterexamples and clarifying the relationship between spectral sequence degeneration and DR-decomposability.

Contribution

It provides the first examples of DR-indecomposable special fibers in semi-stable reductions over Witt rings, answering an open problem by Illusie.

Findings

Counterexamples to DR-decomposability in semi-stable reductions

E_1 degeneration does not imply DR-decomposability

Clarification of the relationship between spectral sequence degeneration and DR-decomposability

Abstract

We answer negatively an open problem of Illusie on the DR-decomposability of the log de Rham complex of the special fiber of a semi-stable reduction over the Witt ring. We also show that $E_1$ degeneration of the Hodge to log de Rham spectral sequence does not imply DR-decomposability of semi-stable varieties.