# On $E_1$-degeneration for the special fiber of a semistable family

## TL;DR

This paper investigates the invariance of the $E_1$-degeneration property of the logarithmic Hodge to de Rham spectral sequence in semistable families, showing it depends only on the generic fiber and providing explicit examples and algebraic proofs.

## Contribution

It proves the $E_1$-degeneration is invariant under admissible blow-ups and depends solely on the generic fiber, while also showing decomposability is not invariant, addressing an open problem.

## Key findings

- $E_1$-degeneration is invariant under admissible blow-ups
- Dependence of $E_1$-degeneration on the generic fiber
- Explicit examples showing non-invariance of decomposability

## Abstract

We study the $E_1$-degeneration of the logarithmic Hodge to de Rham spectral sequence of the special fiber of a semistable family over a discrete valuation ring. On the one hand, we prove that the $E_1$-degeneration property is invariant under admissible blow-ups. Assuming functorial resolution of singularities over $\mathbb{Z}$, this implies that the $E_1$-degeneration property depends only on the generic fiber. On the other hand, we show by explicit examples that the decomposability of the logarithmic de Rham complex is not invariant under admissible blow-ups, which answer negatively an open problem of L. Illusie (Problem 7.14 \cite{Illusie2002}). We also give an algebraic proof of an $E_1$-degeneration result in characteristic zero due to Steenbrink and Kawamata-Namikawa.