A reformulation of the Hodge conjecture in terms of derived algebraic cycles - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:09:37Z http://mathoverflow.net/feeds/question/108962 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108962/a-reformulation-of-the-hodge-conjecture-in-terms-of-derived-algebraic-cycles A reformulation of the Hodge conjecture in terms of derived algebraic cycles lagrangiansubmanifold 2012-10-05T21:30:12Z 2012-10-05T22:29:30Z <p>This question is inspired partly by an earlier MathOverflow question, in particular, see <a href="http://mathoverflow.net/questions/30396/derived-algebraic-geometry-and-chow-rings-chow-motives" rel="nofollow">http://mathoverflow.net/questions/30396/derived-algebraic-geometry-and-chow-rings-chow-motives</a></p> <p>Two interesting possibilities were posed, that could lead to an equivalent formulation of the Hodge conjecture:</p> <p>Given the following condition is true, we could reformulate a 'derived' version of the Hodge conjecture (see this <a href="http://mathoverflow.net/questions/30396/derived-algebraic-geometry-and-chow-rings-chow-motives/36371#36371" rel="nofollow">answer of Peter Scholze</a> )</p> <blockquote> <p>1) Any 'derived algebraic cycle' gives rise to virtual fundamental classes in all cohomology theories, which again are Hodge or Tate cycles.</p> </blockquote> <p>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. </p> <p>A good reference for the notion of the Chow ring is the following:</p> <p><a href="http://www.math.jussieu.fr/~voisin/Articlesweb/weyllectures.pdf" rel="nofollow">http://www.math.jussieu.fr/~voisin/Articlesweb/weyllectures.pdf</a></p> <p>So, my question is </p> <p><strong>Question</strong> What progress has been made regarding 1)?</p> <p>I greatly appreciate any answers. </p>