Abstract. We study algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH 0 (X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the Griffiths' intermediate Jacobian J 2 (X) when the ground field is C. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove representability of zero-cycles on several classes of threefolds fibred by surfaces with algebraic H 2 . This gives another new examples of three-dimensional varieties whose motives are finite-dimensional.
We study symmetric powers in the homotopy categories of abstract closed symmetric monoidal model categories, in both unstable and stable settings. As an outcome, we prove that symmetric powers preserve the Nisnevich and étale homotopy type in the unstable and stable motivic homotopy theories of schemes over a base. More precisely, if f is a weak equivalence of motivic spaces, or a weak equivalence between positively cofibrant motivic spectra, with respect to the Nisnevich or étale topology, then all symmetric powers Sym n (f ) are weak equivalences too. This gives left derived symmetric powers in the corresponding motivic homotopy categories of schemes over a base, which aggregate into a categorical λ-structures on these categories.
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