2 typo

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 cohomoloycohomology.

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.

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 cohomoloy.

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.