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

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.