PHD COURSE / CORSO DI DOTTORATO
                                                                                                                  
                                                                                                   (ANNO ACCADEMICO 2012/2013)




                      TOROIDAL COMPACTIFICATIONS OF LOCALLY SYMMETRIC VARIETIES 

                                                                                                                               DOCENTE: Filippo Viviani

DIARY OF LECTURES: 15/03/2013: Hermitian symmetric manifolds (HSMs) and their decomposition into euclidean, compact and non-compact type. 22/03/2013: Lie groups and Lie algebras associated to (non-euclidean) HSMs. Real forms of complex semisimple Lie algebras with involutions: duality between compact forms and Cartan decompositions. Duality between (irreducible) HSMs of compact and non-compact type. 05/04/2013: HSMs of non-compact type as bounded symmetric domains: Bergmann metric and Harish-Chandra embedding. Classification of irreducible HSMs. Borel embeddings of a HSM of non-compact type into its compact dual.   12/04/2013: HSMs of compact type as cominuscle homogeneous rational projective varieties. Arithmetic groups and their action on bounded symmetric domains. Locally symmetric varieties.   19/04/2013: Modular forms. Baily-Borel-Satake compactification and its minimality property.   03/05/2013: Boundary components. Classification of boundary components via parabolic subgroups. Rational boundary components. The Baily-Borel-Satake compactification as a quotient of the rational closure endowed with its Satake topology. The boundary of the Baily-Borel-Satake compactification. 09/05/2013: The structure of the normalizer of a boundary component: the 5-term decomposition. The decomposition of a bounded symmetric domain along a boundary component: projections onto the boundary and projection onto its associated symmetric cone. 10/05/2013: Homogenous, self-adjoint and symmetric cones. Jordan algebras. Semisimple vs Simple Jordan algebras. 17/05/2013: Euclidean Jordan algebras. The correspondence between (simple) Euclidean Jordan algebras, complex (simple) semisimple Jordan algebras and (irreducible) symmetric cones. The classification theorem. 17/05/2013 (bis): Tube domains associated to symmetric cones. 24/05/2013: Some key results for Euclidean Jordan algebras: Jordan frames, spectral theorem, Peirce decomposition. The boundary of a symmetric cone. Rational closure. Polyehdral decompositions. 07/05/2013: Review of toric varieties: toric varieties and fans; morphisms of toric varieties; geometric properties: smoothness, completeness, projectivity. 07/05/2013 (bis): Toroidal compactifications of locally symmetric varities associated to an admissible polyhedral decomposition. Geometric properties: smoothness and projectivity. Applications: Kodaira dimension of locally symmetric varieties of sufficiently high level; Kodaira dimension of the moduli space of abelian varieties and K3 surfaces.

INTRODUCTION


Locally symmetric (Hermitian) spaces are quotients of bounded symmetric domains (or equivalently non-compact Hermitian symmetric domains) by arithmetic subgroups of their automorphism group.

The study of locally symmetric spaces lies at the intersection of many fields of mathematics, such as: algebraic geometry, differential geometry, number theory, representation theory, etc..

In algebraic geometry, locally symmetric spaces often occur naturally as moduli spaces, e.g.
the moduli space of elliptic curves (with torsion points, level structures, etc.);
the moduli space of polarized abelian varieties (with endomorphism structures, level structures, etc..);
the moduli space of polarized K3 surfaces;
the moduli space of Enriques surfaces;
some configuration spaces of points.

Locally symmetric spaces are quasi-projective varieties but they are not complete. The problem of compactify them is therefore very natural and important.

Baily-Borel showed how to compactify locally symmetric varieties by taking the Proj of a suitable ring of modular forms. The resulting compactification (called the minimal
or Baily-Borel compactification) is however highly singular and therefore not suited for many purposes.

Smooth compactifications of locally symmetry varieties were later constructed by Ash-Mumford-Rapoport-Tai and called toroidal compactifications. These compactifications depends
on the choice of an admissible family of rational polyehedral decomposition of some homegeneous self-adjoint cones. An essential role in their construction is played by the theory of
toroidal embeddings due to Kempf-Knudsen-Mumford-Saint-Donat.

 

PROGRAM


The course aims at describing the construction and main properties of the toroidal compactifications of locally symmetric varieties.

The topics that will be covered are the following:
(1) Bounded symmetric domains and locally symmetric varieties;
(2) Homogeneous self-adjoint cones and their polyhedral decompositions;
(3) Toroidal embeddings;
(4) Minimal (or Baily-Borel) compactifications.
(5) Toroidal compactifications.

Throughout the course, special emphasis will be put on the following guiding examples:
the Siegel space, the moduli space of abelian varieties, the cone of positive definite quadratic forms.



REFERENCES


         Basic References

• A. Ash, D. Mumford, M. Rapoport, Y.S. Tai: Smooth Compactifications of Locally Symmetric Varieties. Cambridge University Press, 2nd edition, 2010.
• Y. Namikawa: Toroidal compactification of Siegel spaces. Lecture Notes in Mathematics Vol. 812, Springer-Verlag, 1980.
• G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat: Toroidal embeddings I . Lecture Notes in Mathematics Vol. 339, Springer-Verlag, 1973.
  

   
           Auxiliary References

• A. Borel, L. Ji: Compactifications of symmetric and locally symmetric spaces. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 2006.
• C. L. Chai: Compactification of Siegel moduli schemes. London Mathematical Society Lecture Note Series, 107. Cambridge University Press, Cambridge, 1985.
• G. Faltings, C. L. Chai: Degeneration of abelian varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 22. Springer-Verlag, Berlin, 1990. 
• J. S. Milne: Introduction to Shimura varieties. Notes available at http://www.jmilne.org/math/articles/2005aX.pdf.

I expect to be able to write some notes of the course that will be distributed to the participants.




PREREQUISITES

Basic knowledges of algebraic geometry.
Some knowledge of toric geometry and algebraic groups/Lie algebras could be useful (although not strictly necessary).



 
FOR ANY INFORMATION OR QUESTIONS: Please contact me at viviani(at)mat.uniroma3.it