Course Outline
My goal is to give an introduction to the
BlochBeilinson conjectures and their
consequences for complex projective varieties.
I would like to present several filtrations
which are natural candidates
to be the BlochBeilinson filtrations on
Chow groups, and to compare them.
I would also like to split
as much as possible the BlochBeilinson
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.
 Basic of Hodge theory, Chow groups,
cycle class. Correspondences, and the
generalized Mumford theorem.
 The BlochBeilinson conjecture on
the existence of a filtration: description
of several candidates.
 Spread out cycles and Beilinson's conjecture
for varieties over
a number field. Nori's theorem and applications; relations
with variations of Hodge structures.
 Filtrations on the Griffiths group.
A generalized Nori conjecture and its relation
with the BlochBeilinson conjecture.
 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=c_{1}(O_{X}(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 BlochBeilinson
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.
