On p-adic vanishing cycles of log smooth families

TL;DR

This paper demonstrates that the sheaf of p-adic vanishing cycles in log smooth families over a DVR of mixed characteristic is generated by Milnor symbols, utilizing Kato's computation on graded quotients.

Contribution

It establishes a new understanding of the structure of p-adic vanishing cycles in log smooth families, linking them to Milnor symbols and building on Kato's filtration analysis.

Findings

Sheaf of p-adic vanishing cycles is generated by Milnor symbols.

Utilizes Kato's computation on graded quotients of filtrations.

Provides a structural description of vanishing cycles in mixed characteristic.

Abstract

In this paper, we will show that the sheaf of p-adic vanishing cycles of a log smooth family over a DVR of mixed characteristic is generated by Milnor symboles. A key ingredient is a computation (due to K. Kato) on the graded quotients of a multi-indexed filtration on the sheaf concerned, which has been used in several papers of the first author.