In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a dual abelian variety can be defined from an
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
''A'', defined over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K''.
Definition
To an abelian variety ''A'' over a field ''k'', one associates a dual abelian variety ''A''
v (over the same field), which is the solution to the following
moduli problem
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such s ...
. 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
# for all
, the restriction of ''L'' to ''A''× is a degree 0 line bundle,
# the restriction of ''L'' to ×''T'' is a trivial line bundle (here 0 is the identity of ''A'').
Then there is a variety ''A''
v and a line bundle
,, called the Poincaré bundle, which is a family of degree 0 line bundles parametrized by ''A''
v in the sense of the above definition. 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 functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and ...
s 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
natural isomorphism
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural ...
between the double dual ''A''
vv and ''A'' (defined via the Poincaré bundle) and that it is
contravariant functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
ial, i.e. it associates to all morphisms ''f'': ''A'' → ''B'' dual morphisms ''f''
v: ''B''
v → ''A''
v 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 scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in ...
s of dual abelian varieties are
Cartier duals of each other. This generalizes the
Weil pairing
Weil may refer to:
Places in Germany
*Weil, Bavaria
*Weil am Rhein, Baden-Württemberg
*Weil der Stadt, Baden-Württemberg
*Weil im Schönbuch, Baden-Württemberg
Other uses
* Weil (river), Hesse, Germany
* Weil (surname), including people with ...
for elliptic curves.
History
The theory was first put into a good form when ''K'' was the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. In that case there is a general form of duality between the
Albanese variety
In mathematics, the Albanese variety A(V), named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.
Precise statement
The Albanese variety is the abelian variety A generated by a variety V taking a given point of V to ...
of a
complete variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism
:X \times Y \to Y
is a closed map (i.e. maps closed sets onto closed sets). Thi ...
''V'', and its
Picard variety
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global v ...
; this was realised, for definitions in terms of
complex tori
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...
, as soon as
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
had given a general definition of Albanese variety. For an abelian variety ''A'', the Albanese variety is ''A'' itself, so the dual should be ''Pic''
0(''A''), the
connected component of the identity element of what in contemporary terminology is the
Picard scheme
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global v ...
.
For the case of the
Jacobian variety
In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian vari ...
''J'' of a
compact Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
''C'', the choice of a
principal polarization
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular fun ...
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
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.
Theorem
Let the Taylor series
G (x) = \sum_^\infty a_k x^k
be a powe ...
. For general abelian varieties, still over the complex numbers, ''A'' is in the same
isogeny In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel.
If the groups are abelian varieties, then any morphism of the underlyi ...
class as its dual. An explicit isogeny can be constructed by use of an
invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
''L'' on ''A'' (i.e. in this case a
holomorphic line bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a com ...
), 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 ''Â'' extends to any field ''K'' of
characteristic zero
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
. 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
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...
. The definition of ''K''(''L'') has to be in terms of a
group scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in ...
that is a scheme-theoretic
stabilizer, and the quotient taken is now a quotient by a subgroup scheme.
Dual isogeny (elliptic curve case)
Given an
isogeny In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel.
If the groups are abelian varieties, then any morphism of the underlyi ...
:
of
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s of degree
, the dual isogeny is an isogeny
:
of the same degree such that
:
Here