In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field k. A 1-dimensional abelian variety is an elliptic curve, and every elliptic curve is isomorphic to its dual, but this fails for higher-dimensional abelian varieties, so the concept of dual becomes more interesting in higher dimensions.
[1] To A one then associates a dual abelian variety Av (over the same field), which is the solution to the following moduli problem.
, called the Poincaré bundle, which is a family of degree 0 line bundles parametrized by Av in the sense of the above definition.
[2] Moreover, this family is universal, that is, to any family L parametrized by T is associated a unique morphism f: T → Av so that L is isomorphic to the pullback of P along the morphism 1A×f: A×T → A×Av.
Applying this to the case when T is a point, we see that the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety.
In the language of representable functors one can state the above result as follows.
The contravariant functor, which associates to each k-variety T the set of families of degree 0 line bundles parametrised by T and to each k-morphism f: T → T' the mapping induced by the pullback with f, is representable.
The universal element representing this functor is the pair (Av, P).
This association is a duality in the sense that there is a natural isomorphism between the double dual Avv and A (defined via the Poincaré bundle) and that it is contravariant functorial, i.e. it associates to all morphisms f: A → B dual morphisms fv: Bv → Av in a compatible way.
This generalizes the Weil pairing for elliptic curves.
The theory was first put into a good form when K was the field of complex numbers.
For an abelian variety A, the Albanese variety is A itself, so the dual should be Pic0(A), the connected component of the identity element 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.
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 of translations on L that take L into an isomorphic copy is itself finite.
In that case, the quotient is isomorphic to the dual abelian variety Av.
This construction of Av extends to any field K of characteristic zero.
[3] In terms of this definition, the Poincaré bundle, a universal line bundle can be defined on The construction when K has characteristic p uses scheme theory.
is the same as giving a family of degree zero line bundles on
This is then the required family of degree zero line bundles on
One can show using this description that this map is an isogeny of the same degree as
[5] Hence, we obtain a contravariant endofunctor on the category of abelian varieties which squares to the identity.
[6] A celebrated theorem of Mukai[7] states that there is an isomorphism of derived categories
denotes the bounded derived category of coherent sheaves on X.
Historically, this was the first use of the Fourier-Mukai transform and shows that the bounded derived category cannot necessarily distinguish non-isomorphic varieties.
is a complex of coherent sheaves, we define the Fourier-Mukai transform
is exact on the level of coherent sheaves, and in applications
is often a line bundle so one may usually leave the left derived functors underived in the above expression.
Note also that one can analogously define a Fourier-Mukai transform
using the same kernel, by just interchanging the projection maps in the formula.