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.

中文翻译：

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半稳定族特殊纤维的$E_1$-退化

我们研究了离散评估环上半稳定族特殊纤维的对数 Hodge 到 de Rham 谱序列的 $E_1$-退化。一方面，我们证明了 $E_1$-degeneration 性质在允许的爆炸下是不变的。假设奇点在 $\mathbb{Z}$ 上的函子解析，这意味着 $E_1$-degeneration 属性仅依赖于泛型纤维。另一方面，我们通过明确的例子表明，对数 de Rham 复数的可分解性在允许的爆炸下不是不变的，这否定地回答了 L. Illusie 的一个开放问题（问题 7.14 \cite{Illusie2002}）。由于 Steenbrink 和 Kawamata-Namikawa，我们还给出了特征零中 $E_1$-退化结果的代数证明。