Timeline for A lift of the second Chern class
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 30, 2015 at 6:54 | vote | accept | Alex Gavrilov | ||
| Nov 30, 2015 at 5:23 | comment | added | Dan Petersen | @PavelSafronov Thanks. To be perfectly honest, I have no clue whether there is a higher algebraic gerbe lying around... | |
| Nov 29, 2015 at 21:02 | comment | added | Pavel Safronov | @DanPetersen, thanks for your comment. All I meant to say is that there is no algebraic 2-gerbe lying around since the cycle map uses analytic topology (correct me if I'm wrong). | |
| Nov 29, 2015 at 20:42 | comment | added | Dan Petersen | @PavelSafronov Your last sentence is a bit weird - why do you say that the Chern classes in Chow are "totally different" for $n \geq 2$? The cycle map from Chow ring to Deligne cohomology is compatible with the Chern classes for all $n$. Do you just mean that the map $\mathrm{CH}^n(X) \to H^{2n}_{\mathcal D}(X,\mathbf Z(n))$ is not an isomorphism in general for $n \geq 2$? | |
| Nov 29, 2015 at 12:19 | vote | accept | Alex Gavrilov | ||
| Nov 29, 2015 at 12:21 | |||||
| Nov 29, 2015 at 12:18 | comment | added | Alex Gavrilov | Well, it looks simple enough once you know the stuff. | |
| Nov 29, 2015 at 10:58 | comment | added | Pavel Safronov | Deligne cohomology is defined in Section 1 of wwwmath.uni-muenster.de/u/pschnei/publ/beilinson-volume/…, Section 8 talks about Chern classes. Using the quasi-isomorphism $(\mathbf{Z}\rightarrow\mathcal{O})\cong \mathcal{O}^\times[-1]$ you get the hypercohomology group I mentioned. | |
| Nov 29, 2015 at 10:51 | comment | added | Alex Gavrilov | It is good, but I want a reference for this. | |
| Nov 29, 2015 at 10:50 | comment | added | Pavel Safronov | The Deligne cohomology group $H^{2p}_D(X, \mathbf{Z}(p))$ is isomorphic to $H^{2p-1}(X, \mathcal{O}^\times\rightarrow \Omega^1\rightarrow ...\rightarrow \Omega^{p-1})$ which projects to $H^{2p-1}(X, \mathcal{O}^\times)$. | |
| Nov 29, 2015 at 10:42 | comment | added | Alex Gavrilov | In Deligne cohomology projects to $H^3(X,O^∗)$, but how? Do you mean that there is a map $H^{2p}_D(X,Z(p))\to H^{2p-1}(X,O^*)$? | |
| Nov 29, 2015 at 10:39 | comment | added | Alex Gavrilov | Naturally, I mean topological Chern classes. You say that the Chern class | |
| Nov 29, 2015 at 10:17 | history | answered | Pavel Safronov | CC BY-SA 3.0 |