Connectivity and Purity for logarithmic motives

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 (P1,∞)-local complexes of sheaves with log transfers. The homotopy t-structure on logDMeff(k) is proved to be compatible with Voevodsky’s t-structure; that is, we show that the comparison functor R□¯¯¯¯ω∗:DMeff(k)→logDMeff(k) is t-exact. The heart of the homotopy t-structure on logDMeff(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