I am trying to understand the following statement from Gillet book. If someone could provide me reference for that. "The hodge conjecture can be understood as assertions about the image of the natural transformation from algebraic to topological K-theory."
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1$\begingroup$ The relevant piece of technology (at least rationally) is the Chern character isomorphism between topological/algebraic K-theory (tensor Q) and a direct sum of cohomology/Chow groups (tensor Q). Perhaps there is more that can be said integrally, but I am kind of skeptical... $\endgroup$– dhyCommented Aug 6, 2018 at 3:54
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$\begingroup$ Can hodge conjecture be formulated using K-theory ? I think as you mentioned Chern character and probably Atiyah-Hirzebruch spectral sequence. We will have a spectral sequence from cohomology to topological K-theory. If Hodge conjecture is true then it means that it would take algebraic cycles into something sensible on the topological K-theory part ? $\endgroup$– user115794Commented Aug 6, 2018 at 4:08
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$\begingroup$ Atiyah-Hirzebruch is unnecessary. The point is that Hodge is about the image of Chow -> cohomology, and so translates immediately via these direct sum decompositions (which are compatible with the algebraic $K_0$ to topological $K_0$ map.) Explicitly, the topological Chern isomorphism gives you a Hodge structure on rationalized topological K-theory (or rather, a direct sum of Hodge structures of different weights), and the middle piece is again the algebraic part. $\endgroup$– dhyCommented Aug 6, 2018 at 4:11
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$\begingroup$ Can you explain more in details ? I don't understand what you mean by the fact that the middle piece is algebraic ? $\endgroup$– user115794Commented Aug 6, 2018 at 17:48
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