Timeline for For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?
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13 events
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| Dec 19, 2013 at 17:57 | vote | accept | Qiaochu Yuan | ||
| Dec 10, 2013 at 18:43 | history | edited | Donu Arapura | CC BY-SA 3.0 |
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| Dec 10, 2013 at 14:09 | history | edited | Donu Arapura | CC BY-SA 3.0 |
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| Dec 10, 2013 at 13:54 | comment | added | Dan Petersen | @DonuArapura Actually I think your original statement was OK, you just need a further conjecture! If you assume both the Hodge conjecture and the Bloch-Beilinson conjectures on filtrations of Chow groups, then the cycle class map from Chow to cohomology is an isomorphism (with $\mathbf Q$ coefficients) if and only if all off-diagonal Hodge numbers vanish. | |
| Dec 10, 2013 at 8:38 | comment | added | Daniel Litt | Not to pile on, but I also don't see why injectivity holds for $0$-cycles in your surface example. This seems to imply Bloch's conjecture for surfaces with $h^1(X)=0$, which is open to my knowledge. | |
| Dec 10, 2013 at 7:50 | history | edited | Donu Arapura | CC BY-SA 3.0 |
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| Dec 10, 2013 at 7:49 | comment | added | Donu Arapura | Dan, you're right of course! In my defense, it's past 1am here. | |
| Dec 10, 2013 at 7:46 | comment | added | Dan Petersen | I don't understand your last statement. Wouldn't the Hodge conjecture just imply surjectivity of the cycle class map? | |
| Dec 10, 2013 at 7:44 | comment | added | Piotr Achinger | Is the Hodge conjecture not known even in the case $h^{pq} = 0$ for $p\neq q$, i.e. when every class is a Hodge class? | |
| Dec 10, 2013 at 7:40 | comment | added | Piotr Achinger | About the necessity of tensoring with $\mathbb{Q}$ in the last statement - Kollar's example: on a very general degree $48$ hypersurface in $\mathbb{P}^4$, every curve has even degree, and $h^{p, q} = 0$ for $p\neq q$, $p+q \neq 3$. | |
| Dec 10, 2013 at 7:36 | comment | added | Qiaochu Yuan | Thanks for the response! My reference for the claim that the cellular result is due to Totaro is Eisenbud and Harris' 3264 and All That, in a comment below Proposition 1.19, although admittedly this is still a draft. | |
| Dec 10, 2013 at 7:25 | history | edited | Donu Arapura | CC BY-SA 3.0 |
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| Dec 10, 2013 at 7:18 | history | answered | Donu Arapura | CC BY-SA 3.0 |