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.
Let A be an abelian variety over a field k. We define
\operatorname{Pic}0(A)\subset\operatorname{Pic}(A)
m*L\congp*L ⊗ q*L
m,p,q
A x kA\toA
\operatorname{Pic}0(A)
To A one then associates a dual abelian variety Av (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrized by a k-variety T is defined to be a line bundle L on A×T such that
t\inT
Then there is a variety Av and a line bundle
P\toA x A\vee
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. The n-torsion of an abelian variety and the n-torsion of its dual are dual to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group schemes of dual abelian varieties are Cartier duals of each other. 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. 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 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. 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.[3] 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.[4]
Let
f:A\toB
f
\hat{f}:\hat{B}\to\hat{A}
\hat{A}
\hat{f}:\hat{B}\to\hat{A}
A
\hat{B}
To this end, consider the isogeny
f x 1\hat{B
(f x 1\hat{B
PB
B
A
By the aforementioned functorial description, there is then a morphism
\hat{f}:\hat{B}\to\hat{A}
(\hat{f} x
*P | |
1 | |
A |
\cong(f x 1\hat{B
f
\hat{\hat{f}}=f
Hence, we obtain a contravariant endofunctor on the category of abelian varieties which squares to the identity. This kind of functor is often called a dualizing functor.[6]
Db(A)\congDb(\hat{A})
Db(X)
Recall that if X and Y are varieties, and
l{K}\inDb(X x Y)
X\toY | |
\Phi | |
l{K |
X\toY | |
\Phi | |
l{K |
Note that
p
p*
l{K}
\Phil{K
The statement of Mukai's theorem is then as follows.
Theorem: Let A be an abelian variety of dimension g and
PA
A x \hat{A}
\hat{A | |
\Phi | |
PA |
\toA}\circ
A\to\hat{A | |
\Phi | |
PA |
\iota:A\toA
[-g]
A\to\hat{A | |
\Phi | |
PA |
. David Mumford. Abelian Varieties. Oxford University Press. 978-0-19-560528-0. 2nd. 1985.