Some computations in the heart of the homotopy t-structure on logarithmic motives

TL;DR

Problem: show that the forgetful functor from log motives to Nisnevich transfers is fully faithful and that $h_0^{\overline{\square}}$ is $\mathbf{A}^1$-invariant, without relying on resolution of singularities. Approach: develop a general method to compute motivic localization $h_0^{\overline{\square}}$ and determine the first homotopy of $M^{\mathrm{eff}}(\mathbf{P}^1)$ via Gysin triangles, purity, and residue theory; establish $h_0^{\overline{\square}}(\mathbf{P}^1,\mathrm{triv})=\mathbb{Z}$ and $h_1^{\overline{\square}}(\mathbf{P}^1,\mathrm{triv})\simeq \omega^*\mathbf{G}_m$, aligning with $\omega^*h_1^{A^1}(\mathbf{P}^1)$. Contributions: prove that the obstruction to fullness vanishes in the heart $\mathbf{logCI}^{\mathrm{ltr}}(k,\mathbb{Z})$, deducing that $\omega_\sharp \iota$ is fully faithful in the transfers setting (and resolving BM22 Conjecture 0.2 for transfers); this also corrects gaps in prior work. Impact: provides a robust bridge between log motives and Nisnevich transfers and informs future $\mathbf{A}^1$-stable comparisons in log motivic homotopy theory.

Abstract

In this note we will illustrate a method for computing the $π_0$ of the effective log motive of a smooth and proper variety over a perfect field $k$ and show that it is $\A^1$-invariant. We will apply this to compute the first homotopy groups of $¶^1$ to show that the stripping functor from log motivic sheaves to (usual) Nisnevich sheaves with transfers is fully faithful.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Corollary 2.4
• Lemma 2.5
• Proposition 2.6
• Theorem 2.7
• Proposition 3.2
• proof