Timeline for What do you lose when passing to the motive?
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| Mar 1, 2010 at 0:25 | comment | added | JBorger | I'm not sure how confident I am in that point of view yet. But it is similar to the situation in lambda-algebraic geometry, where the Witt vector space of a variety X is accessible through the functor it corepresents, and the dual construction, the arithmetic jet space functor, is accessible through the functor it represents. And indeed, the Witt functor tells you about the motive, by de Rham-Witt theory, and the arithmetic jet space functor tells you about the rational points, following Buium. | |
| Mar 1, 2010 at 0:21 | comment | added | JBorger | @Emerton: From a certain perspective, I think this is not surprising. You could say that motives, being a cohomology theory, are about maps from varieties X into some fixed generalized variety H, which would be some kind of generalization of an Eilenberg-Mac Lane space. But rational points of X are about maps into X. In other words, rational points are about the functor represented by X, whereas the motive of X is about the functor corepresented by X. I think that typically it's hard to go from information about one to the other. | |
| Feb 28, 2010 at 16:28 | comment | added | Qing Liu | In a slightly different context: in characteristic 0, non isomorphic torsors under (possibly distinct) abelian varieties have distinct classes in the Grothendieck ring $K_0(Var_K)$ (which has a specialization map to the $K_0$ of Chow motives). This is a consequence of the following theorem of Larsen & Lunts and of Bittner: over a field $K$ of characteristic $0$, if two projective smooth and geometrically connected varieties $X, Y$ have the same class in $K_0(Var_K)$, then they are stably birational. B. Poonen used this result to construct 0-divisors in $K_0(Var_{\mathbb Q})$. | |
| Feb 28, 2010 at 10:54 | history | edited | Chandan Singh Dalawat | CC BY-SA 2.5 |
Bonus.; edited body
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| Feb 28, 2010 at 8:03 | history | edited | Chandan Singh Dalawat | CC BY-SA 2.5 |
smooth *projective* conics
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| Feb 28, 2010 at 6:32 | comment | added | Emerton | The point about torsors over abelian varieties is an interesting one. For example, it makes it hard to see how one might study Sha of an elliptic curve over $\mathbb Q$ (say) from a Langlandsish point of view, because it's not clear what you might do motivically with an element of Sha (with the hope that whatever you did could then be interpreted automorphicly) which would distinguish it from the elliptic curve itself. | |
| Feb 28, 2010 at 5:13 | comment | added | Evgeny Shinder | Over Q Severi-Brauer varieties and quadrics all have the same motives indeed, but integrally they are different. | |
| Feb 28, 2010 at 4:04 | history | answered | Chandan Singh Dalawat | CC BY-SA 2.5 |