Timeline for Relationship between Dolbeault and de Rham cohomology on Riemann surface
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 22, 2021 at 15:22 | comment | added | Donu Arapura | The argument I had in mind is that $H^1(X,\Omega)\to H^2(X,\mathbb{C})$ is surjective map of vector spaces of the same dimension (by duality), so.... | |
| Jun 22, 2021 at 15:09 | comment | added | Martin Skilleter | I’m much more comfortable with line bundles, so Serre duality is a very acceptable tool for me. In this case, I think the map following $\pi$ in this long exact sequence becomes (after applying Serre duality) the exterior derivative from anti-holomorphic functions into holomorphic 1-forms, so everything is in the kernel. If so, I am fully satisfied. Thank you for all your help. | |
| Jun 22, 2021 at 15:03 | vote | accept | Martin Skilleter | ||
| Jun 22, 2021 at 14:46 | comment | added | Donu Arapura | Right, I didn't elaborate because it wouldn't be trivial. You could use Serre duality, for example, but this is a story in itself. | |
| Jun 22, 2021 at 14:36 | comment | added | Martin Skilleter | I am very happy with this, but my concern is that in proving surjectivity of $\pi$, we secretly need some harmonic analysis, the details of which I cannot prove. The claim that $\pi$ is surjective amounts to the statement that everything is in the kernel of the map $H^1(X, \mathcal{O}_X) \to H^1(X, \Omega^1_X)$ induced by the exterior derivative, and this is not at all clear to me. | |
| Jun 22, 2021 at 14:23 | vote | accept | Martin Skilleter | ||
| Jun 22, 2021 at 14:34 | |||||
| Jun 22, 2021 at 14:08 | history | edited | Donu Arapura | CC BY-SA 4.0 |
added 384 characters in body
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| Jun 22, 2021 at 13:52 | vote | accept | Martin Skilleter | ||
| Jun 22, 2021 at 14:21 | |||||
| Jun 22, 2021 at 13:04 | history | answered | Donu Arapura | CC BY-SA 4.0 |