9
$\begingroup$

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?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ I am not fully sure what Deligne is referring to, but maybe he means absolute Hodge cycles/motivated cycles/etc? $\endgroup$
    – David
    Commented Jun 26, 2014 at 6:33
  • $\begingroup$ If you want to start looking into this, I suggest reading the introduction of Lecture Notes 900, and §§0–4 of Andre's paper “Pour une théorie inconditionnelle des motifs”. (Both references are also mentioned by anon, below.) They are very readable and quickly give you the idea of what is going on. $\endgroup$
    – jmc
    Commented Jun 26, 2014 at 13:37

1 Answer 1

Reset to default
4
$\begingroup$

Deligne's absolute Hodge classes (Deligne 1982 LNM), Andre's motivated classes (Andre 1996 IHES), Milne's rational Tate classes (Milne 2009 Moscow MJ),...

Improve this answer
$\endgroup$
1
  • $\begingroup$ And of course all the variations on absolute Hodge depending on which cohomology theories one takes into account (for example Ogus [in LNM900] add crystalline data+conditions). $\endgroup$
    – jmc
    Commented Jun 26, 2014 at 12:57

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .