A Voevodsky motive associated to a log scheme

TL;DR

This paper constructs a new Voevodsky motive associated with fs log schemes over a field, linking it to log Betti cohomology and exploring applications in motivic neighborhoods and monodromy filtrations.

Contribution

It introduces a geometrical Voevodsky motive for fs log schemes and establishes its relation to log Betti cohomology over complex numbers.

Findings

Betti realization of the motive matches log Betti cohomology

Applications to motivic tubular neighborhoods

Insights into limit motives and monodromy filtrations

Abstract

For each fs log scheme $(X,\mathcal M_X)$ over a field $k$ we construct a geometrical Voevodsky motive $[X]^{log}\in DM_{gm}(k,\mathbb Q)$. We prove that, for $k=\mathbb C$, the Betti realization of $[X]^{log}$ is the log Betti cohomology of $(X, \mathcal M_X)$. We give applications to motivic tubular neighborhoods, limit motives and the monodromy filtrations.