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: $H\_1(C,\; mathbb\{Z\})\; cong\; mathbb\{Z\}^\{2g\}.$Geometrically, this homology group consists of (homology classes of) "cycles" in "C", or in other words, closed loops. Therefore we can choose "2g" loops $gamma\_1,\; dots,\; gamma\_\{2g\}$ generating it. On the other hand, another, more algebro-geometric way of saying that the genus of "C" is "g", is that: $H^0(C,\; K)\; cong\; mathbb\{C\}^g,$ where "K" is the canonical bundle on "C".By definition, this is the space of globally-defined differential forms on "C", so we can choose "g" linearly independent forms $omega\_1,\; dots,\; omega\_g$. Given forms and closed loops we can integrate, and we define "2g" vectors: $Omega\_j\; =\; left(int\_\{gamma\_j\}\; omega\_1,\; dots,\; int\_\{gamma\_j\}\; omega\_g\; ight)\; in\; mathbb\{C\}^g.$It follows from the Riemann bilinear relations that the $Omega\_j$ generate a nondegenerate lattice $Lambda$ (that is, they are a real basis for $mathbb\{C\}^g\; cong\; mathbb\{R\}^\{2g\}$), and the Jacobian is defined by: $J(C)\; =\; mathbb\{C\}^g/Lambda.$

The Abel–Jacobi map is then defined as follows. We pick some base point $p\_0\; in\; C$ and, nearly mimicking the definition of $Lambda$, define the
Although this is seemingly dependent on a path from $p\_0$ to $p,$ any two such paths define a closed loop in $C$ and, therefore, an element of $H\_1(C,\; mathbb\{Z\}),$ so integration over it gives an element of $Lambda.$ Thus the difference is erased in the passage to the quotient by $Lambda.$

Invariant construction of the Abel–Jacobi map

Let $M$ be a manifold. Let $pi=pi\_1(M)$ be its fundamental group. Let $f:\; pi\; o\; pi^\{ab\}$ be its abelianisation map. Let$tor=\; tor(pi^\{ab\})$ be the torsion subgroup of$pi^\{ab\}$. Let $g:\; pi^\{ab\}\; o\; pi^\{ab\}/tor$be the quotient by torsion. Clearly,$pi^\{ab\}/tor$ is non-canonically isomorphic to$mathbb\{Z\}^\{2g\}$, where $g$ is the genus. Let $phi=gcirc\; f\; :\; pi\; o\; mathbb\{Z\}^b$ be the composed homomorphism.

Definition. The cover of the manifold$M$ corresponding the subgroup $mathrm\{Ker\}(phi)subset\; pi$ is called the universal (or maximal) free abeliancover.

Now assume "M" has a Riemannian metric. Let $E$ be the space of harmonic $1$-forms on$M$, with dual $E^*$ canonically identified with$H\_1(M,mathbb\{R\})$. By integrating an integralharmonic $1$-form along paths from a basepoint $x\_0inM$, we obtain a map to the circle$mathbb\{R\}/mathbb\{Z\}=S^1$.

Similarly, in order to define a map $M\; o\; H\_1(M,mathbb\{R\})\; /H\_1(M,mathbb\{Z\})\_\{mathbb\{R$ without choosing a basis forcohomology, we argue as follows. Let $x$ be a point in the
universal cover $ilde\{M\}$ of $M$. Thus$x$ is represented by a point of $M$ togetherwith a path $c$ from $x\_0$ to it. Byintegrating along the path $c$, we obtain a linear form,$h\; o\; int\_c\; h$, on $E$. We thus obtain a map$ilde\{M\}\; o\; E^*\; =\; H\_1(M,mathbb\{R\})$, which,furthermore, descends to a map

:$overline\{A\}\_M:\; overline\{M\}\; o\; E^*,;;\; cmapsto\; left(hmapsto\; int\_c\; h\; ight),$

where $overline\{M\}$ is the universal free abelian cover.

Definition. The Jacobi variety (Jacobi torus) of $M$ is thetorus

:$J\_1(M)=H\_1(M,mathbb\{R\})/H\_1(M,mathbb\{Z\})\_mathbb\{R\}.$

Definition. The Abel–Jacobi map

:$A\_M:\; M\; o\; J\_1(M),$

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.

Abel–Jacobi theorem

The following theorem was proved by Abel and Jacobi (each one proved one implication): Suppose that: $D\; =\; sum\_i\; n\_i\; p\_i$is a divisor (meaning a formal integer-linear combination of points of "C"). We can define: $u(D)\; =\; sum\_i\; n\_i\; u(p\_i)$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 $n\_i$ are all positive integers, then: $u(D)\; =\; u(E)$ if and only if $D$ is linearly equivalent to $E.$

References

* cite book
author = E. Arbarello
coauthors = M. Cornalba, P. Griffiths, J. Harris
title = Geometry of Algebraic Curves, Vol. 1
year = 1985
series = Grundlehren der Mathematischen Wissenschaften
publisher = Springer–Verlag
isbn = 978-0387909974
chapter = 1.3, "Abel's Theorem"