In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety.
In other words, the complex torus J(C), with certain 'markings', is enough to recover C. The same statement holds over any algebraically closed field.
It can be recast to read that a certain natural morphism, the period mapping, from the moduli space of curves of a fixed genus, to a moduli space of abelian varieties, is injective (on geometric points).
Firstly, to geometric questions about that morphism, for example the local Torelli theorem.
A case that has been investigated deeply is for K3 surfaces (by Viktor S. Kulikov, Ilya Pyatetskii-Shapiro, Igor Shafarevich and Fedor Bogomolov)[3] and hyperkähler manifolds (by Misha Verbitsky, Eyal Markman and Daniel Huybrechts).