Triangulated categories of logarithmic motives over a field

TL;DR

This paper develops a new theory of logarithmic motives over fields, using finite log correspondences and a special topology, and shows its relation to existing motivic theories with several fundamental properties established.

Contribution

It introduces a novel framework for motives of logarithmic schemes, incorporating $ar{ox}$-homotopy invariance and connecting to Voevodsky's motives under resolution of singularities.

Findings

Hodge cohomology is $ar{ox}$-invariant and representable in the logarithmic motives category.

Fundamental properties like Mayer-Vietoris, Gysin sequence, and blow-up formulas are established.

The category relates closely to Voevodsky's motives and $A^1$-invariant theories.

Abstract

In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the basic idea of parameterizing homotopies by $\overline{\square}$, i.e. the projective line with respect to its compactifying logarithmic structure at infinity. We show that Hodge cohomology of log schemes is a $\overline{\square}$-invariant theory that is representable in the category of logarithmic motives. Our category is closely related to Voevodsky's category of motives and $\mathbb{A}^{1}$-invariant theories: assuming resolution of singularities, we identify the latter with the full subcategory comprised of $\mathbb{A}^{1}$-local objects in the category of logarithmic motives. Fundamental properties such as $\overline{\square}$-homotopy invariance, Mayer-Vietoris for coverings, the analogs of the Gysin sequence and the Thom space isomorphism as well as a blow-up formula and a projective bundle formula witness the robustness of the setup.