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.
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Construction of the 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. Therefore, we can choose 2g loops
By definition, this is the space of globally defined holomorphic differential forms on C, so we can choose g linearly independent forms
It follows from the Riemann bilinear relations that the
The Abel–Jacobi map is then defined as follows. We pick some base point
Although this is seemingly dependent on a path from
The Abel–Jacobi map of a Riemannian manifold
Let
Definition. The cover
Now assume M has a Riemannian metric. Let
Similarly, in order to define a map
where
Definition. The Jacobi variety (Jacobi torus) of
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. Interestingly, 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.
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
Jacobi proved that this map is also surjective, 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). For higher-dimensional compact projective varieties the Albanese variety and the Picard variety are dual but need not be isomorphic.