**SCHOOL (AND WORKSHOP) ON
**

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

**
**First announcement

**Lecturers.
**

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

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

**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.

**Further announcements.**

A more detailed second announcement (containing informations on
accomodation, registration and financial supports) will follow
probably in February 2009.