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, giobol@mate.polimi.it

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