Professor Olivier Debarre is a leading expert in algebraic geometry. He obtained his State doctor (docteur d’État) in 1987 under the supervision of Arnaud Beauville. He became professor at the Université Louis-Pasteur (Strasbourg I) in 1995. Since 2008, he is professor at the Université Paris Diderot (Paris VII) and serves as the deputy director (directeur adjoint) of the Department of mathematics and their applications at the École normale supérieure.

Time:  from April 30th to May 25th, Mondays 13:00-14:50, Fridays 10:10-12:00.

Abstract: A complex torus of dimension g is the quotient of a complex vector space of dimension g by a lattice (of rank 2g). It has a canonical structure of a compact complex variety. In this course, we will study the geometry of complex tori, the main question being the existence of a holomorphic embedding of the torus in a projective space. When such an embedding exists, we say that the complex torus is an abelian variety and becomes an algebraic object.
Starting with the best-known case of elliptic curves (g=1), we will study classical objects such as theta functions but we will also introduce along the way more modern tools of modern complex algebraic and analytic geometry, such as differential forms, Kähler forms, de Rham cohomology. 
The course will roughly follow the book ``Complex Tori and Abelian Variety,'' written by the instructor, O. Debarre.

Reference: Olivier Debarre, Complex tori and abelian varieties. SMF/AMS Texts and Monographs, 11. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2005. x+109 pp.