Quasi-unipotent motives and motivic nearby sheaves

Let $k$ be an algebraically closed field of characteristic zero. We consider a relative version over a general $k$‑scheme of the category of quasi-unipotent motives introduced by J. Ayoub over $k$. We introduce a monodromic version of the nearby motivic sheaf functor associated with a function $f : X \to \mathrm{A}^1_k$ on a separated $k$‑scheme of finite type and show that the motives obtained by applying it are quasi-unipotent. Using this construction, we prove a comparison between this monodromic version of the motivic nearby sheaf of J. Ayoub and the theory of virtual nearby cycles of J. Denef and F. Loeser that takes into account the monodromy action.

Keywords

motivic sheaves, nearby motivic sheaves, quasi-unipotent motives, motivic Milnor fiber

2010 Mathematics Subject Classification

14C15, 14F42, 32S30

Full Text (PDF format)

Received 8 August 2017

Accepted 28 April 2020

Published 30 September 2021