On the logarithmic slice filtration

We consider slice filtrations in logarithmic motivic homotopy theory. Our main results establish conjectured compatibilities with the Beilinson, BMS, and HKR filtrations on (topological, log) Hochschild homology and related invariants. In the case of perfect fields admitting resolution of singularities, the motivic trace map is compatible with the slice and BMS filtrations, yielding a natural morphism from the motivic Atiyah-Hirzerbruch spectral sequence to the BMS spectral sequence. Finally, we consider the Kummer \'etale hypersheafification of logarithmic $K$-theory and show that its very effective slices compute Lichtenbaum \'etale motivic cohomology.