In
mathematics, the Abel–Jacobi map is a construction of
algebraic geometry which relates an
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
to its
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 var ...
. In
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, it is a more general construction mapping a
manifold to its Jacobi torus.
The name derives from the
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
of
Abel
Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd ...
and
Jacobi that two
effective divisors are
linearly equivalent
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David ...
if and only if they are indistinguishable under the Abel–Jacobi map.
Construction of the map
In
complex algebraic geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
, the Jacobian of a curve ''C'' is constructed using path integration. Namely, suppose ''C'' has
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
''g'', which means topologically that
:
Geometrically, this
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
consists of (homology classes of) ''cycles'' in ''C'', or in other words, closed loops. Therefore, we can choose 2''g'' loops
generating it. On the other hand, another more algebro-geometric way of saying that the genus of ''C'' is ''g'' is that
:
where ''K'' is the
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, it ...
on ''C''.
By definition, this is the space of globally defined holomorphic
differential forms on ''C'', so we can choose ''g'' linearly independent forms
. Given forms and closed loops we can integrate, and we define 2''g'' vectors
:
It follows from the
Riemann bilinear relations that the
generate a nondegenerate
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
(that is, they are a real basis for
), and the Jacobian is defined by
:
The Abel–Jacobi map is then defined as follows. We pick some base point
and, nearly mimicking the definition of
define the map
:
Although this is seemingly dependent on a path from
to
any two such paths define a closed loop in
and, therefore, an element of
so integration over it gives an element of
Thus the difference is erased in the passage to the quotient by
. Changing base-point
does change the map, but only by a translation of the torus.
The Abel–Jacobi map of a Riemannian manifold
Let
be a smooth compact
manifold. Let
be its fundamental group. Let
be its
abelianisation
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
map. Let
be the torsion subgroup of
. Let
be the quotient by torsion. If
is a surface,
is non-canonically isomorphic to
, where
is the genus; more generally,
is non-canonically isomorphic to
, where
is the first Betti number. Let
be the composite homomorphism.
Definition. The cover
of the manifold
corresponding to the subgroup
is called the universal (or maximal) free abelian cover.
Now assume ''M'' has a
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
. Let
be the space of harmonic 1-forms on
, with dual
canonically identified with
. By integrating an integral harmonic 1-form along paths from a basepoint
, we obtain a map to the circle
.
Similarly, in order to define a map
without choosing a basis for cohomology, we argue as follows. Let
be a point in the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
of
. Thus
is represented by a point of
together with a path
from
to it. By integrating along the path
, we obtain a linear form on
:
:
This gives rise a map
:
which, furthermore, descends to a map
:
where
is the universal free abelian cover.
Definition. The Jacobi variety (Jacobi torus) of
is the torus
:
Definition. The ''Abel–Jacobi map''
:
is obtained from the map above by passing to quotients.
The Abel–Jacobi map is unique up to translations of the Jacobi torus. The map has applications in
Systolic geometry. The Abel–Jacobi map of a Riemannian manifold shows up in the large time asymptotics of the heat kernel on a periodic manifold ( and ).
In much the same way, one can define a graph-theoretic analogue of Abel–Jacobi map as a piecewise-linear map from a finite graph into a flat torus (or a Cayley graph associated with a finite abelian group), which is closely related to asymptotic behaviors of random walks on crystal lattices, and can be used for design of crystal structures.
Abel–Jacobi theorem
The following theorem was proved by Abel: Suppose that
:
is a divisor (meaning a formal integer-linear combination of points of ''C''). We can define
:
and therefore speak of the value of the Abel–Jacobi map on divisors. The theorem is then that if ''D'' and ''E'' are two ''effective'' divisors, meaning that the
are all positive integers, then
:
if and only if
is
linearly equivalent
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David ...
to
This implies that the Abel-Jacobi map induces an injective map (of abelian groups) from the space of divisor classes of degree zero to the Jacobian.
Jacobi proved that this map is also surjective, so the two groups are naturally isomorphic.
The Abel–Jacobi theorem implies that 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 compact complex curve (dual of holomorphic 1-forms modulo periods) is isomorphic to its
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 var ...
(divisors of degree 0 modulo equivalence). For higher-dimensional compact projective varieties the Albanese variety and the Picard variety are dual but need not be isomorphic.
References
*
*
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{{DEFAULTSORT:Abel-Jacobi map
Algebraic curves
Riemannian geometry
Niels Henrik Abel