# Are there “motivic” proofs of Weil conjectures in special cases?

This is a question meant as a first step to get into reading more on Weil conjectures and standard conjectures. It is known that the standard conjectures on vanishing of cycles would imply the Weil conjectures. So, are there proofs of Weil conjectures in special cases using partial results on the standard conjectures? If so, which cases, and what are the references?

Background: Borcherds mentions here that Manin proved a few special cases in higher dimensions using motives.

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Of course, there's Serre's Analogues kählériens de certaines conjectures de Weil, Annals 1960, where he deduces an analogue of the Weil-Riemann hypothesis over $\mathbb{C}$ using standard facts from Hodge theory. This is technically not an answer at all, but I thought I'd mention it since I had the (perhaps mistaken) impression that this was partly the inspiration for the standard conjectures.

The other more relevant comment is that one can give an elementary proof of the Weil conjecture for any smooth variety whose Grothendieck motive lies in the tensor category generated by curves. I should explain, especially in light of Minhyong's comments, that this could be understood as shorthand for saying the variety can built up from curves by taking products, taking images, blow ups along centres which of the same type, and so on. Actually, for such varieties, the Frobenius can be seen to act semisimply. I think this open in general. So perhaps there's some value in this.

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If you believe the Tate conjectures, then every motive belongs to the tensor category generated by motives of curves. I recall from somewhere that Grothendieck had the idea to prove the Weil conjectures by covering varieties with products of curves; then Serre found a surface that couldn't be covered rationally by a product of curves (see Grothendieck-Serre correspondence, 31/3/1964). – Tony Scholl Jul 28 '10 at 19:36

Weil's proofs for curves and abelian varieties essentially use special cases of the standard conjectures, and the framework of the standard conjectures (like many other conjectures of a motivic nature) is suggested by trying to generalize the abelian variety case (or if you like, the case of H^1) to general varieties (or if you like, to cohomology in higher degrees).

Deligne's proof for K3 surfaces uses a motivic relation between K3s and abelian varieties (which is most easily seen on the level of Hodge structures) to import the result for abelian varieties into the context of K3 surfaces. This is not so different in spirit to Manin's proof for unirational 3-folds, except that the relationship between the K3 and associated abelian variety (the so-called Kuga-Satake variety) is not quite as transparent.

[Added, in light of the comments by Donu Arapura and Tony Scholl below:] In the K3 example, it would be better to write "a conjectural motivic relation ... (which can be observed rigorously on the level of Hodge structures) ...".

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For the sake of completeness: Here's the MR link for Deligne's proof for K3 surfaces: ams.org/mathscinet-getitem?mr=296076 – Anweshi Jul 28 '10 at 16:12
When you talk about the motivic relation between K3s and abelian varieties, I assume you mean at the level of absolute Hodge cycles. Isn't it unknown in general as to whether it's given by a genuine correspondence? – Donu Arapura Jul 28 '10 at 18:05
Donu: In fact, Deligne explains in the introduction that his (quite miraculous) proof is inspired by motivic considerations, but that he does not construct an identity between motives. Apart from some trivial cases (eg Kummer surfaces) there are no proven identities between K3 motives and abelian variety motives, except in the absolute Hodge category - which doesn't behave well under reduction mod p. It is the rigidity of families that enables Deligne to get reduction mod p to work. One should read his proof in full - even if the result has been superceded by Weil I/II. – Tony Scholl Jul 28 '10 at 19:21
Tony, thanks for shedding some light. I had wondered how Deligne got mod p consequences out of this very transcendental construction. I should look at it (one day...). – Donu Arapura Jul 28 '10 at 19:49

The paper by Manin is MR0258836 Manin, Ju. I. Correspondences, motifs and monoidal transformations. Mat. Sb. (N.S.) 77 (119) 1968 475--507 where he uses motives to prove the Weil conjectures for unirational projective 3-folds.

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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. – Minhyong Kim Jul 28 '10 at 15:38
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. – Minhyong Kim Jul 28 '10 at 15:41
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 ...) – Emerton Jul 28 '10 at 15:47
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. – Minhyong Kim Jul 28 '10 at 15:55
@Emerton: algebraic topology = algebraic geometry? :) – David Hansen Jul 28 '10 at 16:21