On $E_1$-degeneration for the special fiber of a semistable family
On $E_1$-degeneration for the special fiber of a semistable family
复制标题
半稳定族特殊纤维的$E_1$-退化
DOI：
10.4310/cntp.2020.v14.n3.a4
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发表时间：
2019
影响因子：
1.9
通讯作者：
Junchao Shentu
中科院分区：
文献类型：
--
作者：
Mao Sheng;Junchao Shentu
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.
