Timeline for Can Hodge symmetry and invariance of Hodge numbers in smooth families be proven purely algebraically?
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| Dec 14, 2022 at 6:48 | comment | added | jmc | @PiotrAchinger I agree that I wasn't very precise, and that you can always look at a random splitting when you are dealing with a single cohomology group. Let's say that I don't know of any algebraic proof that there is a splitting that is functorial in morphisms of complex algebraic varieties. Or more precisely, I don't know of an algebraic proof/construction of a Weil cohomology theory for smooth projective complex varieties that takes values in pure Hodge structures. | |
| Dec 13, 2022 at 20:46 | comment | added | R. van Dobben de Bruyn | @PiotrAchinger why not compile your comments into an answer? | |
| Dec 13, 2022 at 17:39 | comment | added | Damian Rössler | @R. Van Dobben de Bruyn. Thank you for this explanation! This surely would work. So the issue is to prove hard Lefschetz for algebraic de Rham cohomology, which then would presumably be a consequence of hard Lefschetz for crystalline or rigid cohomology. One could however argue that this kind of proof is not altogether algebraic because it involves a passage to the limit (because it involves $p$-adic numbers). It is not completely finitary, like Deligne-Illusie's proof of the degeneration of the Hodge to de Rham spectral sequence. | |
| Dec 13, 2022 at 16:13 | comment | added | R. van Dobben de Bruyn | @DamianRössler I think that Hodge symmetry in characteristic $0$ should follow from the algebraic proof of Hodge–de Rham degeneration plus Serre duality plus hard Lefschetz for algebraic de Rham cohomology (which preserves the Hodge filtration because the Chern class of a line bundle lands in the correct piece). This is more or less the method used in the work that Piotr cited. | |
| Dec 13, 2022 at 11:28 | comment | added | Damian Rössler | @Piotr Achinger. I don't understand the reference to Fontaine-Messing. There are smooth and projective surfaces over ${\bf F}_p$, which are liftable to ${\bf Z}/p^2{\bf Z}$, and which do not satisfy Hodge symmetry (see Serre, "Sur la topologie...", Symposium Internacional..., 24-53, Mexico (1958). As far as I know, there is no algebraic approach to Hodge symmetry in general. | |
| Dec 13, 2022 at 7:35 | comment | added | Piotr Achinger | @jmc what do you mean by "proof of splitting"? Many abstract splittings exist, but the one from Hodge theory cannot come from algebraic geometry (the conjugate filtration is anti-holomorphic). In p-adic Hodge theory however, there is such a splitting (the Hodge-Tate decomposition), though again it does not exist in families. | |
| Dec 13, 2022 at 7:33 | comment | added | Piotr Achinger | For (2) this indeed follows from Deligne-Illusie, see section 4.1, though the characteristic zero statement is not spelled out there. See also Theoreme 5.5 in Deligne "Theoreme de Lefschetz...". | |
| Dec 13, 2022 at 7:32 | comment | added | jmc | I agree with the comments above. But I don't know of an algebraic proof of the splitting of the Hodge filtration. For this you seem to need analysis (for now). And that splitting is important for many things that fall under the general umbrella of "doing Hodge theory". | |
| Dec 13, 2022 at 7:26 | comment | added | Piotr Achinger | For (1) see Jean-Marc Fontaine and William Messing "p-adic periods and p-adic etale cohomology", section I.4.4. | |
| Dec 13, 2022 at 7:16 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Dec 13, 2022 at 5:52 | comment | added | abx | Yes, there is: Deligne-Illusie, Relèvements modulo $p^2$ et décomposition du complexe de de Rham. Invent. Math. 89 (1987), no. 2, 247–270. | |
| Dec 13, 2022 at 5:27 | history | asked | William C. Newman | CC BY-SA 4.0 |