Class Web Page for Math 808E Section 0101
Algebraic Cycles
Spring 2015

Instructor Information

Course Information

The class

Here is a quick, but not totally accurate, definition of the field of algebraic cycles: It is the area within algebraic geometry that attempts, often without success, to understand the structure of the Chow groups of algebraic varieties. The definition is not totally accurate for two reasons. Firstly, there are several other groups related to the Chow groups which are also studied by algebraic cyclists. The two main examples are higher Chow groups and (higher) K-theory, but there are also more exotic groups like K-cohomology or algebraic cobordism which can enter the picture. The second reason that the above definition is not totally accurate is that every once in a while there is significant progress. Still, it is safe to say that the area has a reputation of being very difficult, and, often, progress on algebraic cycles takes a rather abstract form. Part of the problem, but also part of the fascination with algebraic cycles, is that it is guided by two of the most important (and difficult) conjectures in mathematics: the Hodge conjectures and the Beilinson conjectures.

For most of the class, I will lecture, but I will be away for about a week and a half and then there will be "guest" lectures. Some of the topics I have listed below are really on particular papers, and I hope to convince people in the class to lecture about some of these.

Possible topics

  1. Chow groups and cycle class maps. See Fulton's Intersection theory.
  2. Motives and the standard conjectures
  3. A little bit of Hodge theory
  4. Lefschetz 1-1 Theorem and the Hodge conjecture
  5. Jacobians, Abel-Jacobi map, Griffiths' intermediate Jacobian
  6. Mumford's infinite dimensionsionality result for 0-cycles
  7. Bloch's conjecture, Rojtman's theorem
  8. The Griffiths group and various non-zero elements theoreof
  9. Normal functions
  10. Bloch's higher Chow groups
  11. A tiny bit about K-theory
  12. Results of Kato-Saito on zero cycles over finite fields
  13. Beilinson conjectures
  14. Counterexamples to the integral Hodge conjecture and related subjects. (See Voisin's recent book. )

Caveats

It is going to be impossible to prove everything or even most of the things about all of the topics above. So we will certainly have to skip proofs and we will probably have to skip topics. Niranjan Ramachandran is teaching a K-theory class (Math 808F). So I will not try to talk a lot about K-theory.

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