What is the proper initiation to the theory of motives for a new student of algebraic geometry?

Question

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 =)

Answer 1

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.

Answer 2

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

Answer 3

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!

Answer 4

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.

Answer 5

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.

Answer 6

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

Answer 7

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.

Answer 8

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.