Torelli theorem
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In mathematics, the Torelli theorem is a classical result of algebraic geometry over the complex number field, stating that a non-singular 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.
This result has had many important extensions. 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). Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the local Torelli theorem). Secondly, to other period mappings. A case that has been investigated deeply is for K3 surfaces.
