Log motivic nearby cycles

TL;DR

This paper introduces the log motivic nearby cycles functor, establishing its properties and connections with existing motivic theories, especially in the context of log smooth models and rigid analytic motives.

Contribution

It defines the log motivic nearby cycles functor and proves its compatibility with motives of proper smooth schemes over DVRs, relating it to Ayoub's functor and rigid analytic motives.

Findings

The functor sends motives of proper smooth schemes to motives of boundaries in log smooth models.

In characteristic zero, motives over the standard log point are equivalent to rigid analytic motives.

The functor relates to Ayoub's motivic nearby cycles, unifying different motivic frameworks.

Abstract

We define the log motivic nearby cycles functor. We show that this sends the motive of a proper smooth scheme over the fraction field of a DVR to the motive of the boundary of a log smooth model assuming absolute purity, which is unconditional in the equal characteristic case. In characteristic $0$, we show that the $\infty$-categories of motives over the standard log point and rigid analytic motives are equivalent, and we relate log motivic nearby cycles functor with Ayoub's motivic nearby cycles functor.