- Abel–Jacobi map
In
mathematics , the Abel–Jacobi map is a construction ofalgebraic geometry which relates analgebraic curve to itsJacobian variety . InRiemannian geometry , it is a more general construction mapping amanifold to its Jacobi torus.The name derives from the theorem of Abel and Jacobi that twoeffective divisor s arelinearly 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-defineddifferential form s 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 theRiemann 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
u colon C o J(C), u(p) = left( int_{p_0}^p omega_1, dots, int_{p_0}^p omega_g ight) mod Lambda.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 itsabelianisation map. Lettor= tor(pi^{ab}) be the torsion subgroup ofpi^{ab}. Let g: pi^{ab} o pi^{ab}/torbe the quotient by torsion. Clearly,pi^{ab}/tor is non-canonically isomorphic tomathbb{Z}^{2g}, where g is the genus. Let phi=gcirc f : pi o mathbb{Z}^b be the composed homomorphism.Definition. The cover ar M of the manifoldM 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 onM, with dual E^* canonically identified withH_1(M,mathbb{R}). By integrating an integralharmonic 1-form along paths from a basepoint x_0inM, we obtain a map to the circlemathbb{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. Thusx 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 mapilde{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"
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