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
Famous quotes containing the word notions:
“Your notions of friendship are new to me; I believe every man is born with his quantum, and he cannot give to one without robbing another. I very well know to whom I would give the first place in my friendship, but they are not in the way, I am condemned to another scene, and therefore I distribute it in pennyworths to those about me, and who displease me least, and should do the same to my fellow prisoners if I were condemned to a jail.”
—Jonathan Swift (1667–1745)
“the full analysis of the notions of saying something and understanding what one said inevitably involves a concept which, as I will show in detail, essentially corresponds to the Cartesian idea of thought.”
—Zeno Vendler (b. 1921)