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What can one say about the Chow group of zero-cycles (up to rational equivalence) for a product of smooth projective varieties and Chow motives (so, I am interested in the kernel $Chow_0(P)\otimes Chow_0(Q)\to Chow_0(P\times Q)$)? I am interested in Chow groups with rational, torsion and $l$-adic coefficients; for rational coefficients I consider motives over universal domains (say, over complex numbbers), but for $l$-adic ones I am interested in arbitrary (perfect) base fields.

I would like to treat certain smash-nilpotens questions. So, I am interested in Chow groups of high powers of $P$ for $\dim P>1$ and also in Chow groups of the product of a large number of curves whose genus is bounded by some constant.

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  • $\begingroup$ You might want to look at the papers W. Raskind, M. Spiess: Milnor K-groups and zero-cycles on products of curves over $p$-adic fields. Compositio Math. 121 (2000), 1-33 and B. Kahn, T. Yamazaki: Somekawa's K-groups and Voevodsky's Hom-groups. $\endgroup$
    – Matthias Wendt
    Commented Oct 5, 2014 at 9:35
  • $\begingroup$ Ok, I see, you are interested in higher dimensions. I guess there are two things that the papers mentioned in my previous comment imply: 1) one might hope that some version of the results works for products of abelian varieties, and 2) it seems already quite difficult to get results for products of curves, so I would expect it to be unlikely that general results exist for arbitrary dimensions. $\endgroup$
    – Matthias Wendt
    Commented Oct 5, 2014 at 10:13
  • $\begingroup$ Thank you!! I will certainly have a look at the papers you mention anyway. $\endgroup$
    – Mikhail Bondarko
    Commented Oct 5, 2014 at 10:19
  • $\begingroup$ Actually, I have realized that it would be ok to compute the Albanese kernel for the product of a large number of curves with genus bounded by some constant. So, I probably need a certain vanishing result for the corresponding Somekawa's K-groups. $\endgroup$
    – Mikhail Bondarko
    Commented Oct 5, 2014 at 13:02

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