Timeline for Is there a proof of Hodge theory using condensed mathematics?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Sep 14, 2022 at 8:20 | comment | added | Gabriel | @Z.M I was imprecise on purpose! I would be happy to learn about any progress whatsoever on this direction :) | |
| Sep 14, 2022 at 8:17 | comment | added | Gabriel | Dear @PeterScholze, if you could expand this a little into an answer, I would be glad to accept it! | |
| Sep 13, 2022 at 18:15 | comment | added | Peter Scholze | Perfect question! This is a problem that's very much on our minds, but where we feel that we do not yet have the correct approach. (In p-adic Hodge theory, the Fargues-Fontaine curve plays a central role, and similarly in complex Hodge theory it seems that the twistor-$\mathbb P^1$ plays an important role (work of Simpson, Mochizuki, ...), and ideally we'd like to develop Hodge theory in a way that makes the twistor-$\mathbb P^1$ appear organically. But we don't yet see how.) | |
| Sep 13, 2022 at 17:24 | comment | added | Z. M | It might be better to formulate the decomposition theorem, to make clear what we start with (a complex smooth projective variety or a complex manifold with some conditions), and the precise conclusion (e.g. simply the existence of a splitting, or induced by something). An example is Deligne–Illusie's proof, which is algebraic but the conclusion seems to be weaker than the decomposition theorem that complex geometers refer to. | |
| Sep 13, 2022 at 7:56 | history | asked | Gabriel | CC BY-SA 4.0 |