Abel–Jacobi map

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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

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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

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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   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 (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 (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.

The Abel–Jacobi map of a compact Riemann surface

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We provide an analytic construction of the Abel-Jacobi map on compact Riemann surfaces.

Let   denotes a compact Riemann surface of genus  . Let   be a canonical homology basis on  , and   the dual basis for  , which is a   dimensional complex vector space consists of holomorphic differential forms. Dual basis we mean  , for  . We can form a symmetric matrix whose entries are  , for  . Let   be the lattice generated by the  -columns of the   matrix whose entries consists of   for   where  . We call   the Jacobian variety of   which is a compact, commutative  -dimensional complex Lie group.

We can define a map   by choosing a point   and setting   which is a well-defined holomorphic mapping with rank 1 (maximal rank). Then we can naturally extend this to a mapping of divisor classes;

If we denote   the divisor class group of   then define a map   by setting  

Note that if   then this map is independent of the choice of the base point so we can define the base point independent map   where   denotes the divisors of degree zero of  .

The below Abel's theorem show that the kernel of the map   is precisely the subgroup of principal divisors. Together with the Jacobi inversion problem, we can say that   is isomorphic as a group to the group of divisors of degree zero modulo its subgroup of principal divisors.

Abel–Jacobi theorem

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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). 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 (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). 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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  • E. Arbarello; M. Cornalba; P. Griffiths; J. Harris (1985). "1.3, Abel's Theorem". Geometry of Algebraic Curves, Vol. 1. Grundlehren der Mathematischen Wissenschaften. Springer-Verlag. ISBN 978-0-387-90997-4.
  • Kotani, Motoko; Sunada, Toshikazu (2000), "Albanese maps and an off diagonal long time asymptotic for the heat kernel", Comm. Math. Phys., 209: 633–670, Bibcode:2000CMaPh.209..633K, doi:10.1007/s002200050033
  • Sunada, Toshikazu (2012), "Lecture on topological crystallography", Japan. J. Math., 7: 1–39, doi:10.1007/s11537-012-1144-4
  • Farkas, Hershel M; Kra, Irwin (23 December 1991), Riemann surfaces, New York: Springer, ISBN 978-0387977034