**SCHOOL (AND WORKSHOP) ON
**

**HODGE THEORY AND ALGEBRAIC
GEOMETRY
**TRENTO, AUGUST 31-SEPTEMBER 5,
2009

**
**Third announcement

**Lecturers.
**

E. Looijenga(Universiteit Utrecht, Nl)C. Voisin(CNRS and IHES – Fr).

Supporting lecturer and tutor.

**General informations on the School and Workshop.
**The School/Workshop is organized by G. Casnati, A.J. Di Scala, R.
Notari, S. Salamon. For contacting the organizers send a mail
to

The School/Workshop is supported by CIRM -Fondazione Bruno Kessler
(formerly CIRM-ITC), by GNSAGA-INdAM, by Dipartimento di
Matematica-Politecnico di Torino and by the joint Project
"Geometria differenziale e analisi globale"
cofinanced by Italian MIUR.**
**The School and the Workshop will take place at

**Aim of the School.
**
The School is mainly aimed to Phd students and young researchers
in Algebraic Geometry, introducing the participants to research,
beginning from a basic level with a view towards the applications and
to the most recent results.

From the historical and scientific viewpoint, Hodge theory as developed by Griffiths and Deligne since the 70's is a powerful tool for studying algebraic varieties in characteristic zero. It consists mostly in the study of the (mixed) Hodge structures associated to algebraic varieties, and their variations. The so-called period map allows one to study moduli spaces qualitatively and the natural local systems of cohomology on them. Sometimes it even gives a uniformization of the relevant moduli spaces. Another aspect of Hodge theory is the study of the Hodge conjecture and its variants (generalized Hodge conjecture, variational Hodge conjecture). These two aspects are related via the study of Hodge loci, which are natural subloci of moduli spaces.

The whole subject presents a mixture of topology, complex geometry, Lie group theory and of course algebraic geometry, which is rather fascinating. The best example of this is the fact that the constant local systems underlying variations of Hodge structures are of a topological nature, while the variation of Hodge structure itself and the Hodge bundles can be defined inside algebraic geometry. Hodge theory is increasingly used in the context of Differential Geometry for the calculation of Dolbeault cohomology of manifolds with special structures generalizing Kähler geometry.

**Aim of Workshop.
**The Workshop is intended to discuss the state of the art in the
different aspects of Hodge theory. Up to now the following speakers
have
confirmed their participation: F. Campana, K. O'Grady, G.P. Pirola, B.
van Geemen, G. Sankaran. People interested in delivering a short
communication are kindly requested to submit the title and an
abstract within August 17

**Schedule.
**The School will start on Monday, 31

**Registration.**

__Participants who do not require financial support__ are expected
to fill in the on line registration
form before August, 17^{th}.