“Algebraic Geometry, D-modules and Foliations”
14 - 19 July, 2008 - Buenos Aires, Argentina
Universidad de Buenos Aires
Facultad de Ciencias Exactas y Naturales
Building: Pabellón 1.
Room: Aula 8 or Aula 9. Administration: Aula 14.
How to get there
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).
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.
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
Monday 14/July/08 - Friday 18/July/08
Course Prof. Schapira
09:00 to 10:30
Course Prof. Ciliberto
11:00 to 12:30
Course Prof. Lins Neto
14:30 to 16:00
Course Prof. Huybrechts
17:00 to 18:30