Timeline for Beilinson-Hodge conjecture and generation of cohomology ring by $H^1$
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Aug 15, 2022 at 6:26 | comment | added | user127776 | @JasonStarr You are right what I had in my mind was not projective varieties. I was considering algebraic closure of function fields. It is related to another question. So the idea was if singular cohomology of algebraic closure of varieties are generated by $H^1$ does it imply the Hodge conjecture? or Beilinson-Hodge? It seems the answer might be positive for Hodge (not sure about Beilinson-Hodge) but then again I don't know if my other question has any chance of having a positive answer or not. | |
| Aug 15, 2022 at 0:45 | comment | added | Jason Starr | I think the second assumption you write in your comment fails for all projective schemes of positive dimension. | |
| Aug 13, 2022 at 18:56 | comment | added | user127776 | @abx You are right, one needs more than just assuming the generation by $H^1$. I think for $i=2j$ if one adds another assumption $\Gamma_H(H^2(X, \mathbb{Q}(1)))=0$ then it implies Hodge. | |
| Aug 13, 2022 at 7:14 | comment | added | abx | Your hypothesis holds for abelian varieties, but the Hodge conjecture is not known for them, already in dimension 4. | |
| Aug 13, 2022 at 6:11 | history | asked | user127776 | CC BY-SA 4.0 |