Vol. 2, No. 2, 2020

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On $p$-adic vanishing cycles of log smooth families

Shuji Saito and Kanetomo Sato

Vol. 2 (2020), No. 2, 309–335
Abstract

In this paper, we will prove that the sheaf of p-adic vanishing cycles on a regular log smooth family is generated by Milnor symbols, assuming that the base dvr contains a primitive p-th root of unity. Our result generalizes the surjectivity results of Bloch and Kato (Inst. Hautes Études Sci. Publ. Math. 63 (1986), 107–152) and Hyodo (Invent. Math. 91:3 (1988), 543–557) to a regular log smooth case.

Keywords
$p$-adic vanishing cycles, Milnor $K$-groups, log smooth families
Mathematical Subject Classification 2010
Primary: 14F20
Secondary: 11G45, 14F30
Milestones
Received: 7 September 2018
Revised: 7 April 2019
Accepted: 22 April 2019
Published: 2 August 2019
Authors
Shuji Saito
Department of Mathematical Sciences
University of Tokyo
Japan
Kanetomo Sato
Department of Mathematics
Chuo University
Tokyo
Japan