Triangulated categories of logarithmic motives over a field

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 no- tion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the basic idea of parameterizing homotopies by □, i.e., the projective line with respect to its compactifying logarithmic structure at infinity. We show that Hodge cohomology of log schemes is a □-invariant theory that is representable in the category of loga- rithmic motives. Our category is closely related to Voevodsky’s category of motives and A1-invariant theories: assuming resolution of singularities, we identify the latter with the full subcategory comprised of A1-local objects in the category of logarithmic motives. Fundamental properties such as □-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.