**SCHOOL**

**“Algebraic Geometry, D-modules and Foliations”**

14 - 19 July, 2008 - Buenos Aires, Argentina

**Location**

Universidad de Buenos Aires

Facultad de Ciencias Exactas y Naturales

Ciudad Universitaria

Building: Pabellón 1.

Room: Aula 8 or Aula 9. Administration: Aula 14.

How to get there

**Courses**

1) *Geometry of projective varieties and degenerations.*

Professor Ciro Ciliberto (Universitá di Roma II, Italy).

2) *Fourier-Mukai transforms.*

Professor Daniel Huybrechts (University of Bonn, Germany).

3) *From D-modules to quantization-deformation modules.*

Professor Pierre Schapira (Institut de Mathématiques de Jussieu, Paris, France).

4) *Spaces of algebraic foliations of codimension one.*

Professor Alcides Lins Neto (IMPA, Río de Janeiro, Brazil).

**Programs:**

1) Secant varieties and
defective varieties. Tangential varieties. Classification of
varieties with extremal tangential properties. Degenerations of
varieties and of maps. Toric degenerations and applications to the
study of secant varieties and interpolation problems. Toric
degenerations and tropical varieties.

Lecture notes: Francesco Russo, Geometry of special varieties.
Russo.pdf

2) The course covers basic as
well as more advanced aspects of the theory of Fourier--Mukai
transforms for algebraic varieties and, more generally, of derived
categories. In particular, we will cover the classification of
algebraic varieties up to derived equivalence in the known cases
(abelian varieties, K3 surfaces, Calabi--Yau varieties, Fano
varieties, varieties of general type, etc). The course will present
the central techniques (Hochschild cohomology, necessary and
sufficient conditions for equivalences) and open questions. If time
permits, we shall talk about t-structures, stability conditions (a la
Bridgeland), the group of autoequivalences and might touch upon
aspects related to mirror symmetry.

References:

Robin Hartshorne, Algebraic Geometry (Springer Verlag, 1977)

Daniel Huybrechts, Fourier-Mukai transforms in Algebraic Geometry (Oxford Mathematical Monographs, 2006).

3) Schapira Program.pdf

Lecture notes: Pierre Schapira, From D-modules to deformation quantization modules.
Schapira Notes

4) The focus of the course will
be the problem of classifying irreducible components of the
spaces of algebraic foliations of codimension one on complex
projective spaces of dimension n>2. This problem was completely
solved for foliations of degree zero, one and two, and it
remains open for higher degrees. On the other hand, some irreducible
components are known in general: components of pull-back type
(linear and non-linear), some components consisting of
foliations with rational first integral, components defined by
logarithmic differential forms, and the so-called Exceptional
components, parametrizing foliations containing n-1 foliations
of dimension one. Along the course we will expose the known results
and the techniques involved in their proof.

Lecture notes: Alcides Lins Neto, Spaces of algebraic foliations of codimension one.
LinsNeto.pdf

**Time-table:**

Monday 14/July/08 - Friday 18/July/08

Course Prof. Schapira

09:00 to 10:30

Coffee break

Course Prof. Ciliberto

11:00 to 12:30

Lunch

Course Prof. Lins Neto

14:30 to 16:00

Coffee break

Course Prof. Huybrechts

17:00 to 18:30