Timeline for Are there "motivic" proofs of Weil conjectures in special cases?
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| Jul 28, 2010 at 17:18 | history | edited | Richard Borcherds | CC BY-SA 2.5 |
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| Jul 28, 2010 at 16:58 | comment | added | Minhyong Kim | As to my motivation, it's not so easy to say. But in the proof under discussion, note that it can be easily given with no reference whatsoever to the category of motives. This is far from one of those situations where one goes through all kinds of convolutions to do away with some supposedly fancy machinery. I would even go so far as to say that it could be presented in a page or so, along the lines described above. Clearly, 'using motives' has to mean more than this. By the way, I think the expository material of the paper is quite nice. It's just not so relevant to the proof itself. | |
| Jul 28, 2010 at 16:43 | comment | added | Emerton | Dear David, In the second usage, yes! I.e. "properties that are more special to algebraic topology ..." should read "properties that are more special to algebraic geometry ...". (Thanks for this!) | |
| Jul 28, 2010 at 16:21 | comment | added | David Hansen | @Emerton: algebraic topology = algebraic geometry? :) | |
| Jul 28, 2010 at 15:55 | comment | added | Minhyong Kim | I agree with Matthew in a philosphical sense. Even though they're unnecessary mathematically, the paper is inspired by the following two motivic ideas (in the notation I used above): (1)$Y$ is a direct summand of $X$; (2)$X$ is the sum of Tate motives and Tate twists of curves. | |
| Jul 28, 2010 at 15:55 | comment | added | Emerton | If I can guess at Minhyong's motivations, the challenges he is referring to might be the following: to apply deeper geometric properties of motives (properties that are more special to algebraic topology, such as Galois representations, Hodge theory and p-adic Hodge theory), and to epxloit the categorical structure in a more serious way. (If you like, Manin's computation is not categorical in the sense that I mean here: it is just computing an Euler characteristic.) | |
| Jul 28, 2010 at 15:47 | comment | added | Emerton | I agree with Minhyong's hard line stance, but want to add a comenstatory remark: the sense in which Manin uses motives, and the reason they appear in the title of his paper, is that he is using the idea that, since the Weil conjectures are essentially homological in nature, one can study them using the idea of cutting up spaces (in this case, cutting up a unirational 3-fold into pieces related to curves and projective spaces). This was always a basic idea of algebraic topology, but the idea of using it in algebraic geometry is part of the yoga of motives. (cont'd ...) | |
| Jul 28, 2010 at 15:41 | comment | added | Minhyong Kim | I realize this is something of a hard line to take, but it seems to be mathematically more fruitful to be somewhat stricter when speaking of 'using motives' in a proof. At least, this stance sets up more interesting challenges to people working on motives. | |
| Jul 28, 2010 at 15:38 | comment | added | Minhyong Kim | It's a nice paper. But I have to disagree that Manin 'uses' motives in any real sense. What he uses are: A unirational three-fold $Y$ admits a generically finite proper map from a variety $X$ that is obtained from $P^3$ using a sequence of blow-up along smooth centers that are curves or points. From the computation of the cohomology of such a blow-up, $X$ satisfies the Weil conjecture. (Because, projective spaces and curves do.) Since the cohomology of $Y$ embeds into that of $X$, $Y$ satifies the Weil conjecture. | |
| Jul 28, 2010 at 15:27 | history | answered | Richard Borcherds | CC BY-SA 2.5 |