The Torelli theorem: compactification and tropicalization

The classical Torelli map is the modular map from the moduli space of smooth projective curves of genus g into the moduli space of principally polarized abelian varieties of dimension g, sending a curve into its Jacobian. The Torelli theorem asserts that the Torelli map is injective on geometric points. We propose two extensions of the Torelli theorem: one for the compactified Torelli map and the other for the tropical Torelli map. The compactified Torelli map was constructed by Alexeev: it is a modular map from the Deligne-Mumford moduli space of stable curves to the Alexeev moduli space of stable semi-abelic pairs, sending a stable curve into its compactified Picard variety of degree g-1, endowed with its natural theta divisor and the action of the generalized Jacobian. In a joint work with L. Caporaso, we give a complete description of the fibers of the compactified Torelli map. On the other hand, in a joint work with S. Brannetti and M. Melo, we construct moduli spaces of tropical curves and tropical abelian varieties and a tropical Torelli map between them. In another joint work with L. Caporaso, we describe the fibers of the tropical Torelli map. I will report on the above two Torelli-type theorems, trying to enlight the relations between them.