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, we show that the slice filtration realizes the BMS filtration on the p-completed topological cyclic homology.Furthermore, the motivic trace map is compatible with the slice and BMS filtrations, yielding a natural morphism from the motivic slice spectral sequence to the BMS spectral sequence.Finally, we consider the Kummer tale hypersheafification of logarithmic K-theory and show that its very effective slices compute Lichtenbaum tale motivic cohomology.14A21, 14F30, 14F42; 19E20 1. Introduction 2653 2. Slices for logarithmic motives 2660 3. Slice filtration on K-theory 2664 4. HKR filtration on logarithmic Hochschild homology 2666 5. BMS filtration on logarithmic topological Hochschild homology 2672 6.The characteristic-p case 2680 7. Very effective slices of Kummer tale K-theory 2685