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.