Dual Abelian Variety - History

History

The theory was first put into a good form when K was the field of complex numbers. In that case there is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety; this was realised, for definitions in terms of complex tori, as soon as André Weil had given a general definition of Albanese variety. For an abelian variety A, the Albanese variety is A itself, so the dual should be Pic0(A), the connected component of what in contemporary terminology is the Picard scheme.

For the case of the Jacobian variety J of a compact Riemann surface C, the choice of a principal polarization of J gives rise to an identification of J with its own Picard variety. This in a sense is just a consequence of Abel's theorem. For general abelian varieties, still over the complex numbers, A is in the same isogeny class as its dual. An explicit isogeny can be constructed by use of an invertible sheaf L on A (i.e. in this case a holomorphic line bundle), when the subgroup

K(L)

of translations on L that take L into an isomorphic copy is itself finite. In that case, the quotient

A/K(L)

is isomorphic to the dual abelian variety Â.

This construction of  extends to any field K of characteristic zero. In terms of this definition, the Poincaré bundle, a universal line bundle can be defined on

A × Â.

The construction when K has characteristic p uses scheme theory. The definition of K(L) has to be in terms of a group scheme that is a scheme-theoretic stabilizer, and the quotient taken is now a quotient by a subgroup scheme.