220210 Carolina Tamborini

Riemannian symmetric spaces are Riemannian manifolds with special symmetry properties. They are important in various fields of geometry. In the seminar, we will be interested in the following fact: the moduli space A_g of principally polarized abelian varieties is a quotient of the Siegel space, which is a Riemannian symmetric space. This fact can be used to approach the study of the Torelli locus, which is the closure in A_g of the image of the moduli space M_g of smooth, complex algebraic curves of genus g via the Torelli map j: M_g-->A_g.

We will first introduce Riemannian symmetric spaces and their totally geodesic submanifolds. Next, we will describe the problem of studying the geometry of the Torelli locus in A_g and its relation with totally geodesic submanifolds of the Siegel space. Finally, we will explain how this is linked to a famous conjecture from Coleman and Oort.