Jacobian variety

In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is a principally polarized abelian variety, of dimension g, and hence, over the complex numbers, it is a complex torus. If p is a point of C, then the curve C can be mapped to a subvariety of J with the given point p mapping to the identity of J, and C generates J as a group.

Over the complex numbers, it can be realized as the quotient space V/L, where V is the dual of the vector space of all holomorphic differentials on C and L is the lattice of all elements of V of the form

where γ is a closed path in C.

The Jacobian of a curve over an arbitrary field was constructed by Weil (1948) as part of his proof of the Riemann hypothesis for curves over a finite field.

The Abel-Jacobi theorem states that the Jacobian of a curve can be identified with its Picard variety of degree 0 divisors modulo linear equivalence. 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.

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.

References

• P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. pp. 333–363. ISBN 0-471-05059-8. 
• J.S. Milne (1986). "Jacobian Varieties". Arithmetic Geometry. New York: Springer-Verlag. pp. pp. 167-212. ISBN 0-387-96311-1. 
• Mumford, David (1975), Curves and their Jacobians, The University of Michigan Press, Ann Arbor, Mich., MR0419430 
• Shokurov, V.V. (2001), "Jacobi variety", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Weil, André (1948), Variétés abéliennes et courbes algébriques, Paris: Hermann, MR0029522, OCLC 826112 

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