Logarithmic motivic homotopy theory

Abstract

This work is dedicated to the construction of a new motivic homotopy theory for (log) schemes, generalizing Morel-Voevodsky's (un)stable $\mathbb{A}^1$-homotopy category. Our framework can be used to represent log topological Hochschild and cyclic homology, as well as algebraic $K$-theory of regular schemes. Additionally, we can realize the cyclotomic trace as a morphism between motivic spectra. Among our applications, we provide a generalized framework of oriented cohomology theories that enables us to produce new residue sequences for (topological) Hochschild, periodic, and cyclic homology of classical schemes. We also compute $THH$ and its variants for Grassmannians, and we define a new version of algebraic cobordism. Finally, we give a construction of a log \'etale stable realization functor, as well as a Kato-Nakayama realization functor, which is of independent interest for applications in log geometry.