In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety.
In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus.
The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.
In complex algebraic geometry, the Jacobian of a curve C is constructed using path integration.
Namely, suppose C has genus g, which means topologically that Geometrically, this homology group consists of (homology classes of) cycles in C, or in other words, closed loops.
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 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 2g vectors It follows from the Riemann bilinear relations that the
define the map Although this is seemingly dependent on a path from
any two such paths define a closed loop in
Thus the difference is erased in the passage to the quotient by
does change the map, but only by a translation of the torus.
is called the universal (or maximal) free abelian cover.
Similarly, in order to define a map
without choosing a basis for cohomology, we argue as follows.
is the universal free abelian cover.
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 (Kotani & Sunada (2000) and Sunada (2012)).
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.
We provide an analytic construction of the Abel-Jacobi map on compact Riemann surfaces.
denotes a compact Riemann surface of genus
dimensional complex vector space consists of holomorphic differential forms.
We can form a symmetric matrix whose entries are
Then we can naturally extend this to a mapping of divisor classes; If we denote
then this map is independent of the choice of the base point so we can define the base point independent map
The below Abel's theorem show that the kernel of the map
The following theorem was proved by Abel (known as Abel's theorem): Suppose that is a divisor (meaning a formal integer-linear combination of points of C).
The theorem is then that if D and E are two effective divisors, meaning that the
are all positive integers, then Jacobi proved that this map is also surjective (known as Jacobi inversion problem), so the two groups are naturally isomorphic.
The Abel–Jacobi theorem implies that the Albanese variety of a compact complex curve (dual of holomorphic 1-forms modulo periods) is isomorphic to its Jacobian variety (divisors of degree 0 modulo equivalence).