Timeline for Chow motive of a fibration
Current License: CC BY-SA 3.0
9 events
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| Jul 15, 2014 at 8:07 | comment | added | Mikhail Bondarko | Still, it is also conjectured that the functor from Chow motives to homological one is conservative (or at least, homologically trivial endomorphisms of Chow motives should be nilpotent; is this equivalent?). Hence if we have $f:A\to B$ and $g:B\to A$ for Chow motives that are inverse to each other modulo homological equivalence, we obtain that $A$ and $B$ are isomorphic. So, Chow motives are not much different from homological ones in this matter. | |
| Jun 30, 2014 at 12:25 | comment | added | Dan Petersen | @jmc Thanks for the correction, you're right. | |
| Jun 30, 2014 at 10:26 | comment | added | jmc | It may be helpful (or pedantic) to note that the realisation functor is expected to be fully faithful from homological motives to $\mathbb{Q}$-HS. This question is about Chow motives, for which this is not true: all rational cycles that are homologically trivial are killed by the realisation functor (so it is only conjectured to be full). | |
| Jun 30, 2014 at 10:17 | comment | added | Matthias Wendt | Ah, right. I realize now that my comment was a bit silly, sorry. | |
| Jun 30, 2014 at 10:10 | comment | added | Dan Petersen | Suppose X and Y are smooth projective and the Hodge strictures H(X) and H(Y) are isomorphic. This isomorphism is a class in $H(X) \otimes H(Y)^\ast = H(X \times Y)$ which is pure of Tate type, by Hodge conjecture it is class of algebraic cycle and gives morphism of motives. | |
| Jun 30, 2014 at 9:44 | comment | added | Matthias Wendt | No, I don't. I just know of the conjecture that a morphism of motives which induces an isomorphism of Hodge structures is an isomorphism. Could you explain how the Hodge conjecture implies that there is a splitting of $F\to E$ on the level of motives? I certainly agree with you that examples as required in the question are unlikely to exist, I am just unsure which conjectures to use to deduce that... | |
| Jun 30, 2014 at 9:28 | comment | added | Dan Petersen | @Matthias Wendt I don't understand your comment. Do you disagree that the realization functor to Hodge structures is expected to be fully faithful? | |
| Jun 30, 2014 at 9:02 | comment | added | Matthias Wendt | Using the conservativity conjecture maybe a bit more subtle. It is surely ok if you have a morphism $E\to F\times B$ or the other way - conservativity then tells you that if this morphism induces an isomorphism of Hodge structures, it induces an isomorphism of motives. But you will probably not have such a morphism on the scheme level... | |
| Jun 30, 2014 at 8:41 | history | answered | Dan Petersen | CC BY-SA 3.0 |