Dual abelian variety

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.

Definition

Let A be an abelian variety over a field k. We define ${\displaystyle \operatorname {Pic} ^{0}(A)\subset \operatorname {Pic} (A)}$ to be the subgroup consisting of line bundles L such that ${\displaystyle m^{*}L\cong p^{*}L\otimes q^{*}L}$, where ${\displaystyle m,p,q}$ are the multiplication and projection maps ${\displaystyle A\times _{k}A\to A}$ respectively. An element of ${\displaystyle \operatorname {Pic} ^{0}(A)}$ is called a degree 0 line bundle on A.^[1]

To A one then associates a dual abelian variety A^v (over the same field), which is the solution to the following

. 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

1. for all ${\displaystyle t\in T}$, the restriction of L to A×{t} is a degree 0 line bundle,
2. the restriction of L to {0}×T is a trivial line bundle (here 0 is the identity of A).

Then there is a variety A^v and a line bundle ${\displaystyle P\to A\times A^{\vee }}$, called the Poincaré bundle, which is a family of degree 0 line bundles parametrized by A^v 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 → A^v so that L is isomorphic to the pullback of P along the morphism 1_A×f: A×T → A×A^v. Applying this to the case when T is a point, we see that the points of A^v correspond to line bundles of degree 0 on A, so there is a natural group operation on A^v 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 (A^v, P).

This association is a duality in the sense that there is a

for elliptic curves.

History

The theory was first put into a good form when K was the field of

Picard scheme

.

For the case of the

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

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

The Dual Isogeny

Let ${\displaystyle f:A\to B}$ be an isogeny of abelian varieties. (That is, ${\displaystyle f}$ is finite-to-one and surjective.) We will construct an isogeny ${\displaystyle {\hat {f}}:{\hat {B}}\to {\hat {A}}}$ using the functorial description of ${\displaystyle {\hat {A}}}$, which says that the data of a map ${\displaystyle {\hat {f}}:{\hat {B}}\to {\hat {A}}}$ is the same as giving a family of degree zero line bundles on ${\displaystyle A}$, parametrized by ${\displaystyle {\hat {B}}}$.

To this end, consider the isogeny ${\displaystyle f\times 1_{\hat {B}}:A\times {\hat {B}}\to B\times {\hat {B}}}$ and ${\displaystyle (f\times 1_{\hat {B}})^{*}P_{B}}$ where ${\displaystyle P_{B}}$ is the Poincare line bundle for ${\displaystyle B}$. This is then the required family of degree zero line bundles on ${\displaystyle A}$.

By the aforementioned functorial description, there is then a morphism ${\displaystyle {\hat {f}}:{\hat {B}}\to {\hat {A}}}$ so that ${\displaystyle ({\hat {f}}\times 1_{A})^{*}P_{A}\cong (f\times 1_{\hat {B}})^{*}P_{B}}$. One can show using this description that this map is an isogeny of the same degree as ${\displaystyle f}$, and that ${\displaystyle {\hat {\hat {f}}}=f}$.^[5]

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]

Mukai's Theorem

A celebrated theorem of Mukai

${\displaystyle D^{b}(A)\cong D^{b}({\hat {A}})}$, where ${\displaystyle D^{b}(X)}$ denotes the bounded derived category of and shows that the bounded derived category cannot necessarily distinguish non-isomorphic varieties.

Recall that if X and Y are varieties, and ${\displaystyle {\mathcal {K}}\in D^{b}(X\times Y)}$ is a complex of coherent sheaves, we define the Fourier-Mukai transform ${\displaystyle \Phi _{\mathcal {K}}^{X\to Y}:D^{b}(X)\to D^{b}(Y)}$ to be the composition ${\displaystyle \Phi _{\mathcal {K}}^{X\to Y}(\cdot )=Rq_{*}({\mathcal {K}}\otimes _{L}Lp^{*}(\cdot ))}$, where p and q are the projections onto X and Y respectively.

Note that ${\displaystyle p}$ is flat and hence ${\displaystyle p^{*}}$ is exact on the level of coherent sheaves, and in applications ${\displaystyle {\mathcal {K}}}$ 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 ${\displaystyle \Phi _{\mathcal {K}}^{Y\to X}}$ using the same kernel, by just interchanging the projection maps in the formula.

The statement of Mukai's theorem is then as follows.

Theorem: Let A be an abelian variety of dimension g and ${\displaystyle P_{A}}$ the Poincare line bundle on ${\displaystyle A\times {\hat {A}}}$. Then, ${\displaystyle \Phi _{P_{A}}^{{\hat {A}}\to A}\circ \Phi _{P_{A}}^{A\to {\hat {A}}}\cong \iota ^{*}[-g]}$, where ${\displaystyle \iota :A\to A}$ is the inversion map, and ${\displaystyle [-g]}$ is the shift functor. In particular, ${\displaystyle \Phi _{P_{A}}^{A\to {\hat {A}}}}$ is an isomorphism.^[8]

Notes

References

• Milne, James S. (2008). Abelian Varieties (v2.00) (PDF).

• ISBN 978-0-19-560528-0
  .

• Bhatt, Bhargav (2017). Abelian Varieties (PDF).

This article incorporates material from Dual isogeny on

Creative Commons Attribution/Share-Alike License

.