Abel–Jacobi map

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.

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 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 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 generate a nondegenerate lattice (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 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. Let be the space of harmonic -forms on , with dual canonically identified with . By integrating an integral harmonic -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 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 . We thus obtain 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. 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 are all positive integers, then

if and only if is linearly equivalent 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 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.


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