In complex algebraic geometry, the Jacobian of a curveC is constructed using path integration. Namely, suppose C has genusg, which means topologically that Geometrically, this homology group consists 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 glinearly 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 , 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 free abelian cover. Now assume M has a Riemannian metric. 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 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 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. 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, 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. 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 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 is isomorphic to its Jacobian variety. For higher-dimensional compact projective varieties the Albanese variety and the Picard variety are dual but need not be isomorphic.
