The Southwestern Center for Arithmetical Algebraic Geometry

Arizona Winter School 2002
Voisin's Course and Project Descriptions

Course Outline

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.

  1. Basic of Hodge theory, Chow groups, cycle class. Correspondences, and the generalized Mumford theorem.
  2. The Bloch-Beilinson conjecture on the existence of a filtration: description of several candidates.
  3. Spread out cycles and Beilinson's conjecture for varieties over a number field. Nori's theorem and applications; relations with variations of Hodge structures.
  4. Filtrations on the Griffiths group. A generalized Nori conjecture and its relation with the Bloch-Beilinson conjecture.
  5. Polarized Hodge structures and a conjecture on the ring of correspondences. Applications and further questions.

Project

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.