TITLE: Candidates for pure algebraic notion of cycles?
QUESTION [9 upvotes]: In the interview http://www.ems-ph.org/journals/newsletter/pdf/2013-09-89.pdf, Deligne said 

For me, this (Hodge conjecture) is a part of the story of motives, and it is not crucial whether it is true or false. If it is true, that's very good and it solves a large part of the problem of constructing motives in a reasonable way. If one can find another purely algebraic notion of cycles for which the analogue of the Hodge conjecture holds, and there are a number of candidates, this will serve the same purpose, and I would be as happy as if the Hodge conjecture were proved.

I wonder what are those "candidates" mentioned by him above, i.e. some possible pure algebraic notions of cycles?

REPLY [4 votes]: Deligne's absolute Hodge classes (Deligne 1982 LNM), Andre's motivated classes (Andre 1996 IHES), Milne's rational Tate classes (Milne 2009 Moscow MJ),...