Timeline for Betti numbers of Proper nonprojective varieties
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 20, 2016 at 3:17 | comment | added | S. Li | @LMN FYI, Serre constructed example of a surface (over characteristic p) with $H^1_{dR}=1$. | |
| Feb 23, 2013 at 21:19 | vote | accept | LMN | ||
| Feb 23, 2013 at 19:14 | comment | added | anon | @LMN: No, I speak of the $\ell$-adic Betti numbers, defined via $\ell$-adic \'etale cohomology for any prime $\ell \neq p$, with $p$ being the characteristic of the field. Some standard facts are: these numbers are independent of the choice of $\ell$ (for proper smooth varieties), and agree with the classical ones if the variety comes via reduction modulo $p$ from something in characteristic $0$. (They do not, however, always equal the de Rham Betti numbers.) In particular, Junecue's result proves evenness in characteristic $0$ as well. | |
| Feb 23, 2013 at 18:34 | comment | added | Dmitri Panov | Donu, thanks for the reference, I'll have a look (and will try to see if indeed I have an alternative proof :) ). LMN, you are welcome :) | |
| Feb 23, 2013 at 18:19 | comment | added | LMN | anon, sorry to be silly - but since this isn't my area I just want to make sure. When you speaks of betti numbers in characteristic $p$, you referring to the algebraic de Rham complex? | |
| Feb 23, 2013 at 17:10 | comment | added | anon | The evenness of odd Betti numbers is true for proper smooth varieties over any algebraically closed field, including positive characteristic ones (by a recent paper by Junecue Suh). | |
| Feb 23, 2013 at 14:18 | comment | added | Donu Arapura | Dimitri, regarding the evenness of the odd Betti numbers, this has indeed been known. It follows from Prop 5.3 of Deligne, Theoreme de Lefschetz... However, if you do find a proof using weak factorization, it would certainly be nice. | |
| Feb 23, 2013 at 14:06 | history | edited | Dmitri Panov | CC BY-SA 3.0 |
added 32 characters in body
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| Feb 23, 2013 at 13:20 | history | answered | Dmitri Panov | CC BY-SA 3.0 |