On the log motivic stable homotopy groups
Doosung Park
TL;DR
The paper compares the log motivic stable homotopy category $logSH(k)$ with the classical motivic stable homotopy category $SH(k)$ over a perfect field $k$ with resolution of singularities, proving a monoidal adjunction between them and showing the log motivic groups agree with the usual motivic groups: $\\pi^{log}_{p,q}(k) \\cong \\pi_{p,q}(k)$. It develops a robust passage between log smooth schemes and ordinary smooth schemes via an adjunction $\\omega_#: logSH(k) \\leftrightarrows SH(k): \\\omega^*$, and demonstrates that the adjunction preserves colimits and monoidal structures, with the essential image of $\\omega^*$ characterized by $A^1$-local objects. The work also establishes descent properties for log motives through a detailed analysis of Nisnevich squares and admissible blow-ups, enabling a coherent comparison at the level of stable homotopy categories. A key outcome is the construction of the logarithmic K-theory spectrum $\\mathbf{logKGL}=\\omega^*\\mathbf{KGL}$, and a streamlined geometric model for it via Grassmannians; under the adjunction this model corresponds to the classical KGL in $SH(k)$. Overall, the results show that incorporating log structures extends motivic homotopy theory without altering the core stable invariants, while providing new geometric tools for computations.
Abstract
We compare the log motivic stable homotopy category and the usual motivic stable homotopy category over a perfect field admitting resolution of singularities. As a consequence, we show that the log motivic stable homotopy groups are isomorphic to the usual motivic stable homotopy groups.