Timeline for Hodge conjecture and K-theory
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 6, 2018 at 17:48 | comment | added | user115794 | Can you explain more in details ? I don't understand what you mean by the fact that the middle piece is algebraic ? | |
| Aug 6, 2018 at 4:11 | comment | added | dhy | 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. | |
| Aug 6, 2018 at 4:08 | comment | added | user115794 | 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 ? | |
| Aug 6, 2018 at 4:07 | review | Close votes | |||
| Aug 7, 2018 at 9:24 | |||||
| Aug 6, 2018 at 3:54 | comment | added | dhy | 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... | |
| Aug 6, 2018 at 3:45 | review | First posts | |||
| Aug 6, 2018 at 5:07 | |||||
| Aug 6, 2018 at 3:42 | history | asked | user115794 | CC BY-SA 4.0 |