The Torelli morphism and the Torelli theorem

Filippo Viviani (Roma Tre)

Program: The Torelli morphism is the modular map from the moduli space/stack of curves to the moduli space/stack of abelian varieties, sending a curve into its polarized Jacobian. A classical result of G. Torelli asserts the injectivity of the Torelli map, i.e. the fact that a curve can be recovered from its polarized Jacobian. In the first part of the course, we will introduce the Torelli morphism, we will study its local structure and we will give an overview of the different known proofs of the Torelli theorem. In the second part of the course, we will describe the known extensions of the Torelli morphism from the moduli space of stable curves to the toroidal compactifications of the moduli space of abelian varieties and we will investigate the fibers of the extended Torelli morphisms. In the third part of the course, we will describe a tropical analogue of the Torelli map and we will present a tropical Torelli theorem.