*"EAGER is a European Algebraic Geometry Research
Training Network*
*supported by the HPP programme of the European
Commission"*

*Eager Advanced School in Algebraic Geometry*

Eager

Centro Internazionale per la Ricerca Matematica, CIRM (Trento)

Dipartimento di Matematica, Politecnico di Milano

Levico Terme, Trento, Italy

May 15-19, 2002

**Cycles on Varieties**

*Announcement*

*"This school is sponsored by the HPP programme
of the European Commission*
*as a high-lever scientific conference (see
grants below)."*

**Lecturers: Herb CLEMENS (Salt Lake City) and Claire
VOISIN (Paris)**

__Program__

Herb Clemens

will do a short course centering on very concrete examples in cycle
theory. In the first part, the lecturer will give a local characterization
of the Hilbert scheme of a K-trivial threefold X at a smooth curve as a
gradient scheme, that is, as the zero-scheme of df where f is a function
on some finite-dimensional polydisk (whose points are real 2-dimensional
submanifolds of X).

In the second part, the lecturer will discuss bounds (in terms of k,
d, and n) for the existence of k-dimensional subvarieties of low geometric
genus (eg genus 0) on generic hypersurfaces of degree d in complex projective
n-space.

Lecture topics:

1. Example of a Hilbert scheme: Lines on the Fermat quintic threefold;

2. Kuranishi data associated to a C-infinity trivialization;

3. Kuranishi data for a deformations of a pair (manifold, submanifold);

4. Hilbert scheme of curves on a K-trivial threefold as a gradient
scheme;

5. Lifting a low-genus subvariety of a generic hypersurface to a Grassmann
variety;

6. Voisin's integrability condition;

7. Lower bounds on genera of low-genus subvarieties of a generic hypersurface.

Claire Voisin

will discuss the Bloch-Beilinson conjectures on algebraic cycles on
smooth projective varieties. This includes one of the simplest conjectures
for varieties defined over Q, which will be shown to have strong consequences
on the complex case.

The goal is to compare several filtrations on the Chow groups, and
to split the Bloch-Beilinson conjectures into several statements. For example,
an important part of Bloch conjecture could be deduced from the fact that
a certain filtration on the group of cycles modulo algebraic equivalence
is finite.

It will be shown how to exploit the particular features of Hodge structures,
in order to formulate certain subconjectures which might be interesting
and possibly easier to work out.

Lecture topics (tentative plan):

1) Basics in Hodge theory and algebraic cycles, correspondences and
the generalizations of Mumford's theorem.

2) Stating the Bloch-Beilinson conjectures. Construction of filtrations.

3) Some consequences of the conjectures. Other filtrations, comparisons.

4) Splitting the Bloch-Beilinson conjectures into pieces. Evidences
and examples.

5) Some geometric questions.

The background knowledge will be standard texts in Hodge theory (eg
Griffiths-Harris) and if possible Bloch's book on algebraic cycles (Duke
university lectures).

There will be space for junior talks. Participants willing to give
such a talk are invited to send the
title and a short abstract to the local organizer, |

__General information__

Updatings on the school will be available at this web page.

The school will take place in Levico, a resort near Trento (Italy),
at the Grand Hotel Bellavista (http://www.ghbellavista.com).
The arrival of participants will be Tuesday, May 14, 2002. Classes will
be from Wednesday, May 15, 2002, to Sunday, May 19, 2002. The departure
of participants will be on Sunday afternoon.

There is a registration fee of 80 Euros, to be paid cash at the
arrival. The full board accommodation in doubleroom will cost 52
Euro per day per person, in singleroom will cost 58 Euro/day and in a double

single use will cost 65 Euro/day.

Updated information and maps for reaching Levico Terme can be found
at the web page

Other interesting web pages concerning the region

__Grants__

Several grants for young researchers (in the official sense of the
European Community supported activities) coming from member or associated
countries of the European Community (covering accomodation, travel expenses
and/or the registration fee) will be available. People interested in having
a support should send the form before March, 31.

__Reply__

If you are interested in participating, please send the form(s), preferably
by e-mail, to

Augusto Micheletti, CIRM

Istituto Trentino di Cultura

I- 38050 POVO (trento), Italy

fax: Italy+0461 810629

e-mail: michelet@science.unitn.it

or contact the local organizer

Giorgio Bolondi

Dipartimento di Matematica

Politecnico di Milano

giobol@mate.polimi.it