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A preliminary apology is in order: I realize that most of my contributions to this site are in the form of reference requests. I understand that this makes it seem as though I do nothing more than sit around most of the time, soaking in as much advanced mathematics as possible, despite my position as a lowly undergraduate. In all actuality, this couldn't be more accurate; I really do just sit around reading maths most of the time.

Alright, now that I put that out there, I am curious as to where I might find a coherent treatment of the theory of motives; one which is below the level of a professional mathematician and roughly suited for readers of Hartshorne or Eisenbud/Harris's wonderful scheme theory text. That is, I want to understand the discipline which I hear extolled as beautiful and complex by researchers in the field, but which is notoriously abstruse and difficult to learn/understand. I wonder if expositions of the theory of motives are necessarily highly technical, or if it is approachable to the ambitious advanced undergraduate.

Thank you again, MO community, for imparting your wisdom regarding good references. It is very much appreciated =)

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Your question is like asking for a treatment of Deligne's work on mixedness of $\ell$-adic sheaves at the level of Hartshorne, or the Langlands Program at the level of Serre's book on representations of finite groups. Wait until you've mastered several different cohomology theories in algebraic geometry (e.g., coherent cohomology, etale cohomology, topological cohomology) and the relations among them. Then you'll be in position to appreciate the point (and why it doesn't necessarily matter to even have a definition of "motive" for the idea to be useful). –  Boyarsky Jun 20 '10 at 4:59
You can find references in the answers to Ilya's question: mathoverflow.net/questions/2146/whats-the-yoga-of-motives Unfortunately, none of them are aimed at new students of algebraic geometry. –  S. Carnahan Jun 20 '10 at 5:34
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Asking for the moon, in my view. Here are 10 "heuristics" that try to place the theory. NB that many people stop at #1, as if this were enough. None of these points is particularly easy to track in the literature, AFAIK.

  1. An irreducible variety V is going to be treated as a "molecule" in this theory, not an "atom". Motives are in a sense parts of varieties. If you count points over finite fields this can look combinatorial (e.g. Euler's formula for the triangulations of a sphere) but has to go a lot deeper.

  2. In cohomology of a variety of dimension n, the top relevant dimension has to be 2n, for reasons that are easy to see over the complex numbers (real dimension), but in general have to do with topological intuitions, such as ramification taking place in codimension 2.

  3. Grothendieck's big-scale pattern of thought involves defining a whole category at once, and understanding it by means of category-level structures and concepts. The "category of motives" is to be understood, in particular its Hom-sets. These are to be modelled on the idea of algebraic correspondence. So it's morally a category of relations.

  4. Algebraic cycles (i): Generally in homology theory, the modern approach is to start with a very abstract definition and worry later about how to represent a class concretely. Here the opposite approach is useful - algebraic cycles are traced on varieties by combinations of subvarieties.

  5. Algebraic cycles (ii): Algebraic cycles need to be subject to equivalence relations, such as linear equivalence for divisors. There are significant technical issues here (vaguely replacing homotopies).

  6. Algebraic cycles (iii): There is (or may be) a paucity of algebraic cycles. Cf. the Hodge conjecture. In other words we lack existence proofs in general.

  7. Problem-solving (i): Assume enough about a good category of motives and you get a conditional proof of the Weil conjectures.

  8. Problem-solving (ii): Motives can conjecturally account (coarsely, Lie algebra level) for the images of Galois representations on l-adic cohomology.

  9. Top-down view: Motives solve the problem of what would be the "universal Weil cohomology", at least in the best of all possible worlds.

  10. Grothendieck's period conjecture: a concrete out-turn in transcendence theory is the conjectural upper bound for the transcendence degree of the periods of abelian varieties. Motives can "catch" enough algebraic cycles to do this.

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You're probably right--carriage before the horse, if you'll excuse the cliche. However, I am sure that it is understandable for one to want to look at sophisticated constructions and methods used by modern research in a particular subject area to provide motivation. I want to experience the wonderment that I felt when passing from calculus to analysis, or basic number theory/algebra to more advanced material. With a subject so rich as AG, it is more difficult to 'see your endgame' as it were. I am sure that you will understand my enthusiasm here on some level. We were all naive once :) –  lambdafunctor Jun 22 '10 at 4:39
Should really add that the original theory is really not known to work (after 45 years). There are ways round this (absolute Hodge cycles, motivic cohomology), but these are less accessible. –  Charles Matthews Jun 22 '10 at 7:30
Right, I look at things which aren't accessible to me (yet) as a proverbial gauntlet having been thrown down. I have a love for abstruse math, for some reason. But again, I partially just want to see what all the 'fuss' is about with motives (and the work of Voevodsky, Beilinson, et al.)--that is, I realize that it is brilliant from what I can gather, and I'm just looking for the 'why'. –  lambdafunctor Jun 23 '10 at 5:20
Some say motive as in motivation, some say motif as in recurring theme in music. In some concrete cases it is actually not so hard to see what is going on: the category is defined from algebraic correspondences by splitting idempotents (Karoubi construction), which is not so terrible. The explicit examples such as Fermat varieties tend to fall in the motif class, while the broad categorical properties hoped for in the motive class. The flamboyant Nick Katz talk ihes.fr/jsp/site/… shows why these ideas live on. –  Charles Matthews Jun 23 '10 at 7:33
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There is a very friendly introduction to motivic homotopy theory starting even below Hartshorne level: The lecture notes from the Nordfjordeid Summer school on Motivic Homotopy Theory. They consist of three chapters:

1.Topological Prerequisites (Dundas)

2.Algebro-Geometric Prerequisites (Levine)

3.Motivic Homotopy Theory (Voevodsky/Röndigs/Østvær) which is not legally available online but has almost the same content as Voevodsky's ICM talk

Note that it is on the level you asked for and from this source you learn in some detail about one modern approach, but not about classical motives and all the "yoga" and motivation and intuition connected to number theory. The closest to being a coherent source going more into that direction is probably Yves André's book (the link gives you the table of contents), otherwise I just know of scattered notes and articles

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Yes, André's book review: smf4.emath.fr/Publications/Gazette/2007/111/… is IMO a very beautifull introduction including mixed motives, even if some people react allergic to it. –  Thomas Riepe Jun 20 '10 at 13:31
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Manin's old article (the first publication on motives, acc. to the author "an exercise" by Grothendieck) is very readable and beautifully written. Very readable and good too are Kleimann's "Motives" in "Algebraic geometry", Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970), pp. 53--82 , and Demazure's article of that time.

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By far the best introduction I've seen is Milne's introduction Motives-Grothendieck's dream from his webpage. Beware though that there are a lot of things that you ought to know before you can fully appreciate everything in the paper. But even a superficial reading is highly rewarding and motivating!

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Motivating... No pun intended, right? –  lambdafunctor Jun 20 '10 at 17:10
^my shamelessly poor sense of humor aside, I now absolutely love this article! I never knew that this gorgeous theory plays such a fundamental role in things like Birch & Swinnerton-Dyer or the Langlands Program, or so many other unsolved problems. Of course, you are right; I do not understand everything (a forteriori, a great deal) of what is contained therein, but it is certainly highly rewarding and... well, motivating. –  lambdafunctor Jun 20 '10 at 17:15
Actually, the pun was uninteded (but oh how much cooler would I appear if it wasn't!? ;) ) Glad to be of assistance. –  Daniel Larsson Jun 20 '10 at 19:17
Where's the beef? Looks like just definitions and formalism. The philosophy of "thinking motivically" is influential, but until there is a proof of the standard conjectures I don't see what to get excited about (like Goldfeld's strategy before Gross-Zagier). The definition looks too easy to tell us something deep; in what sense is it even a "theory"? Does work of Voevodsky et al. rest on a more sophisticated concept of "motive"? If so, maybe that would be more compelling (on par with etale cohomology, which I know very well). Why is lambdafunctor so excited? Sorry for being a party pooper... –  Boyarsky Jun 21 '10 at 2:15
Well I don't think that I'm any more excited than many in the community get when approaching the methods of Voevodsky et al, or the conjectures of Beilinson, etc. I mean, if there weren't something to get excited about, as it were, then I don't think that so many brilliant mathematicians would spend their time thinking about it. I don't understand all of the mechanisms at play here, mind you. I am just partial because my undergraduate adviser is an A^1 homotopy theorist who has also done a lot of work in motivic cohomology. I just wanted to get to a level where I could understand it better. –  lambdafunctor Jun 22 '10 at 4:35
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The word "motive" has a lot of different (although highly related) meanings. I suggest you go ahead to learn about "pure Chow motives" first, before looking at the more complicated theory of mixed motives.

For motivation, it is necessary to have seen at least one Weil-cohomology theory, so you might want to have a look at the Weil conjectures, too.

For technical stuff, you should know what an abelian category is (and then learn the rest along the way).

Related to the theory of motives are also: K-Theory, (stable) homotopy theory of schemes, intersection theory (Chow groups). If you have an interest in any of these topics, it might be good to look at a treatment that covers the relationsship between this and motives, to give a little bit more motivation.

Since there is no abelian category of mixed motives yet, but instead what "feels like" it's derived category, you might want to learn a little bit about derived categories and triangulated categories before walking to (Voevodsky's theory of) mixed motives.

Of course, there is also the AMS Notices article What is ... a motive? by Barry Mazur.

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Well, this is not specifically motives but I liked reading these notes http://www.math.northwestern.edu/~eric/lectures/zurich/ by Eric Friedlander. These lectures introduce a lot of things you want to know if you are interested in motives.

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I would suggest you: A Pamphlet on Motivic Cohomology, Luca Barbieri-Viale, http://arxiv.org/abs/math/0508147 (or the published version) and/or Bruno Kahn's http://people.math.jussieu.fr/~kahn/preprints/kcag.pdf

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Here is a very understandable introductory article by R. Sujatha. For a beginning student this is good.

In my case, after that article, my next encounter with motives was with the more precise definition of a motive from the initial parts of Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points”. It even sort of defines a mixed motive; in fact it is the only definition of mixed motive that I know.

Read the Mathscinet review and also, Jordan Ellenberg's opinion on this remarkable paper of Deligne. I myself was astonished when I first looked into it and saw how much stuff was contained in it.

Deligne's paper "Formes modulaires et représentations $l$-adiques" proving that the Weil conjectures imply the Ramanujan conjecture, is almost close to the theory of motives even though it does not explicitly mention motives. Here the representations of the absolute Galois group on the étale, or rather on the $\ell$-adic, cohomology is considered. This might give some starting insight into the Galois representations approach to motives.

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