The goal of this article is to extend the work of Voevodsky and Morel on the homotopy t-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel’s connectivity theorem and show a purity statement for $({\mathbf {P}}^1, \infty )$ -local complexes of sheaves with log transfers. The homotopy t-structure on ${\operatorname {\mathbf {logDM}^{eff}}}(k)$ is proved to be compatible with Voevodsky’s t-structure; that is, we show that the comparison functor $R^{{\overline {\square }}}\omega ^*\colon {\operatorname {\mathbf {DM}^{eff}}}(k)\to {\operatorname {\mathbf {logDM}^{eff}}}(k)$ is t-exact. The heart of the homotopy t-structure on ${\operatorname {\mathbf {logDM}^{eff}}}(k)$ is the Grothendieck abelian category of strictly cube-invariant sheaves with log transfers: we use it to build a new version of the category of reciprocity sheaves in the style of Kahn-Saito-Yamazaki and Rülling.

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We prove cancellation theorems for reciprocity sheaves and cube-invariant modulus sheaves with transfers of Kahn-Miyazaki-Saito-Yamazaki. This generalizes a cancellation theorem for A 1 -invariant sheaves with transfers, which was proved by Voevodsky. As an application, we get some new formulas for internal homs of the sheaves Ω i of absolute Kähler differentials.

The proof of [1, Lemma 7.2] contains a gap: the equality $\omega _{\sharp } h_{0}(\Lambda _{\mathrm {ltr}}(\eta ,\mathrm {triv})) = \omega _{\sharp } h_{0}(\omega ^{*}\Lambda _{\mathrm {tr}}(\eta ))$ is false. Indeed one can check that for $X\in \mathbf {Sm}(k)$ proper, $$ \begin{align*} \operatorname{Hom}( \omega_{\sharp} h_{0}(\Lambda_{\mathrm{ltr}} (\eta_{X}, \mathrm{triv})), \mathbf{G}_{a}) \neq \operatorname{Hom}( \omega_{\sharp} h_{0} (\omega^{*} \Lambda_{{\mathrm{tr}}}( \eta_{X})) , \mathbf{G}_{a}), \end{align*} $$ as the left-hand side is $\mathbf {G}_{a}(\eta _{X})$ , whereas the right-hand side is $\mathbf {G}_{a}(X)$ . For now, we can give a proof only of a weaker version of [1, Proposition 7.3]:

We define higher pro-Albanese functors for every effective log motive over a field k of characteristic zero, and we compute them for every smooth log smooth scheme X = (X, ∂X). The result involves an inverse system of the coherent cohomology of the underlying scheme as well as a pro-group scheme Alb log (X) that extends Serre's semiabelian Albanese variety of X − |∂X|. This generalizes the higher Albanese sheaves of Ayoub, Barbieri-Viale and Kahn.