Timeline for When Atiyah class and Chern class coincide?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 10, 2022 at 8:23 | vote | accept | Tom | ||
| Nov 10, 2022 at 8:23 | vote | accept | Tom | ||
| Nov 10, 2022 at 8:23 | |||||
| Nov 4, 2022 at 8:29 | vote | accept | Tom | ||
| Nov 10, 2022 at 8:23 | |||||
| Nov 3, 2022 at 3:46 | comment | added | Tom | For a $\partial\bar\partial$-manifold, the natural map $f:H^{p,q}_{BC}(X)\to H_{\bar\partial}^{p,q}(X)$ is an isomorphism, and there is a natural map $g:H^{p,q}_{BC}(X)\to H^{p+q}_{dR}(X,\mathbb C)$, note that $H^k_{dR}(X,\mathbb C)\cong \bigoplus_{p+q=k}H^{p,q}_{BC}(X)$, $H^{p,q}_{BC}(X)$ can be treated as a subspace of the de Rham cohomology, so $g\circ f^{-1}:H^{p,q}_{\bar\partial}(X)\to H^{p+q}_{dR}(X,\mathbb C)$ maps $H^{p,q}_{\bar\partial}(X)$ to a subspace of $H^{p+q}_{dR}(X,\mathbb C)$. | |
| Nov 2, 2022 at 15:54 | comment | added | David E Speyer | @Tom No, I haven't. Sounds like you know about it, though! | |
| Nov 2, 2022 at 15:27 | comment | added | Tom | Have you considered of $\partial\bar\partial$-manifolds? For this kind of manifolds, Bott-Chern cohomology and Dolbeault cohomology are isomorphic, and the Frölicher spectral sequence degenerates at $E_1$ (see Angella & Tomassini's 2013paper p.73 and Remark 5.21 of DGMS75, Real homotopy theory of Kähler manifolds). So there is $H^k(X,\mathbb C)=\bigoplus_{p+q=k}H^{p,q}_{\bar\partial}(X)=\bigoplus_{p+q=k} H_{BC}^{p,q}(X)$, it seems this kind of manifolds have a chance. | |
| Nov 1, 2022 at 16:09 | comment | added | David E Speyer | Yes, or a little more weakly, I don't know of any other setting in which there is a natural way to treat $H^1(\Omega^1)$ as a subspace of $H^2(\mathbb{C})$. | |
| Nov 1, 2022 at 10:29 | comment | added | Tom | I think this is the kind of answer I'm looking for, so you mean $H^1(\Omega^1)$ can be treated as a subspace of $H^2(X,\mathbb C)$ if and only if $X$ is a compact Kähler manifold? | |
| Oct 31, 2022 at 18:34 | comment | added | Z. M | I guess that they both are realizations of some "motivic Chern class" (but I don't know anything about motivic cohomology for complex manifolds). | |
| Oct 31, 2022 at 18:24 | history | answered | David E Speyer | CC BY-SA 4.0 |