Arizona Winter School 2002
My goal is to give an introduction to the Bloch-Beilinson conjectures and their consequences for complex projective varieties. I would like to present several filtrations which are natural candidates to be the Bloch-Beilinson filtrations on Chow groups, and to compare them. I would also like to split as much as possible the Bloch-Beilinson conjectures for complex varieties into pieces. My goal is also to extract from this set of conjectures as many simple geometric questions as possible which might be easier to work out than the general conjectures.
As for the student project, there are two possibilities.
One is doing research on a subject which looks very simple (but might be difficult as usually are the simplest problems in algebraic cycles).
The problem is the following:
When one has a hypersurface X in projective space, one shows easily that the intersection with the class h=c1(OX(1)) is the zero map when restricted to the group of cycles homologous to 0.
For reasons related to the Lefschetz theorem on hyperplane sections, this should hold also for any complete intersection if one believes the Bloch-Beilinson conjecture. The problem would be to try to prove this.
Another possibility would be to work on a paper. One which I find interesting and of a spirit close to my lectures is a paper by Kimura on the notion of finiteness of Chow groups.