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# Communications in Number Theory and Physics

## Volume 14 (2020)

### Number 3

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

Pages: 555 – 584

DOI: https://dx.doi.org/10.4310/CNTP.2020.v14.n3.a4

#### Authors

#### Abstract

We study the $E_1$-degeneration of the logarithmic Hodge to de Rham spectral sequence of the special fiber of a proper semistable family over a discrete valuation ring. On the one hand, we prove that the $E_1$-degeneration property is invariant under admissible blowups. 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 [3]). This shows that the decomposability of the de Rham complex is strictly stronger than the $E_1$-degeneration of the Hodge to de Rham spectral sequence. We also give an algebraic proof of an $E_1$-degeneration result in characteristic zero due to Steenbrink and Kawamata–Namikawa.

#### Keywords

$\log$ de Rham complex, $\log$ Hodge–de Rham spectral sequence

#### 2010 Mathematics Subject Classification

Primary 14F40. Secondary 14C30.

This work is supported by National Natural Science Foundation of China (Grant No. 11622109, No. 11721101), Chinese Universities Scientific Fund (CUSF) and Anhui Initiative in Quantum Information Technologies (AHY150200). The second author is supported by National Natural Science Foundation of China (Grant No. 11901552).

Received 4 April 2019

Accepted 11 February 2020

Published 13 July 2020