Timeline for Hodge decomposition and degeneration of the spectral sequence
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 4, 2019 at 12:36 | comment | added | Anton Mellit | By the way, if you are satisfied with this argument for elliptic curves, then you can probably upgrade it to arbitrary projective varieties using hard Lefschetz. | |
| Jan 4, 2019 at 12:28 | comment | added | Anton Mellit | Take an elliptic curve $E$ with regular differential 1-form $\omega$. Then the statement of Hodge theory basically says that the class of $\omega$ is not a multiple of the class of $\bar\omega$. Equivalently, the integral $\int_E \omega\wedge\bar\omega$ is not zero. Is there any way to prove that this integral is not zero besides showing that (up to a simple factor) it is strictly positive because it is the volume of the fundamental domain? | |
| Sep 29, 2018 at 22:22 | answer | added | gdb | timeline score: 12 | |
| Sep 29, 2018 at 19:26 | comment | added | Alexander Braverman | Well, the problem is really with the proof, not with the statement - at least for Kahler manifolds you can construct the decomposition intersecting various terms of the Hodge filtration with their complex conjugates. I would like to see a proof of this statement which doesn't use harmonic forms. | |
| Sep 29, 2018 at 19:20 | comment | added | R. van Dobben de Bruyn | My understanding is that the theory of variations of Hodge structures shows that we should not expect the Hodge decomposition to be too canonical (algebraically), otherwise we could do it in families. This suggests at least that there should be some non-algebraic input (but maybe throwing in the complex conjugation is already enough). I would be interested in a more complete and coherent answer. | |
| Sep 29, 2018 at 18:19 | history | asked | Alexander Braverman | CC BY-SA 4.0 |