On $E_1$-degeneration for the special fiber of a semistable family
| Autor: | Junchao Shentu, Mao Sheng |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Mathematics::Commutative Algebra Fiber (mathematics) 14C30 14F40 Open problem Zero (complex analysis) General Physics and Astronomy Resolution of singularities Discrete valuation ring Mathematics - Algebraic Geometry Mathematics::Algebraic Geometry Spectral sequence FOS: Mathematics Algebraic number Invariant (mathematics) Algebraic Geometry (math.AG) Mathematical Physics Mathematics |
| Popis: | 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. 20 pages |
| Databáze: | OpenAIRE |
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