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Mar 9, 2020 at 12:20 comment added Mikhail Bondarko Yes, Jannsen result is not very old; that is why I did not put it on the first place in my answer.:) A terminological remark: I think that it is better to speak about numerical motives than about Chow motives modulo numerical equivalence. Moreover, Chow motives are certainly useful, and one does not realy want to replace them by an abelian category.
Mar 8, 2020 at 21:10 history answered R. van Dobben de Bruyn CC BY-SA 4.0