A reformulation of the Hodge conjecture in terms of derived algebraic cycles

2012-10-05T21:30:12Z

This question is inspired partly by an earlier MathOverflow question, in particular, see http://mathoverflow.net/questions/30396/derived-algebraic-geometry-and-chow-rings-chow-motives

Two interesting possibilities were posed, that could lead to an equivalent formulation of the Hodge conjecture:

Given the following condition is true, we could reformulate a 'derived' version of the Hodge conjecture (see this answer of Peter Scholze )

1) Any 'derived algebraic cycle' gives rise to virtual fundamental classes in all cohomology theories, which again are Hodge or Tate cycles.

In the above sense, a 'derived algebraic cycle' is given by the Chow ring under an equivalence relation, which would then define a multiplication in it.

A good reference for the notion of the Chow ring is the following:

http://www.math.jussieu.fr/~voisin/Articlesweb/weyllectures.pdf

So, my question is

Question What progress has been made regarding 1)?

I greatly appreciate any answers.

A reformulation of the Hodge conjecture in terms of derived algebraic cycles - MathOverflow

A reformulation of the Hodge conjecture in terms of derived algebraic cycles