Further Notions
Torelli's theorem states that a complex curve is determined by its Jacobian (with its polarization).
The Schottky problem asks which principally polarized abelian varieties are the Jacobians of curves.
The Picard variety, the Albanese variety, and intermediate Jacobians are generalizations of the Jacobian for higher dimensional varieties. For varieties of higher dimension the construction of the Jacobian variety as a quotient of the space of holomorphic 1-forms generalizes to give the Albanese variety, but in general this need not be isomorphic to the Picard variety.
Read more about this topic: Jacobian Variety
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