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.

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[edit] 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\right) \in \mathbb{C}^g.

It follows from the Riemann bilinear relations that the Ωj generate a nondegenerate lattice Λ (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 Λ, define the map

u \colon C \to J(C), u(p) = \left( \int_{p_0}^p \omega_1, \dots, \int_{p_0}^p \omega_g\right) \bmod \Lambda.

Although this is seemingly dependent on a path from p0 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 Λ. Thus the difference is erased in the passage to the quotient by Λ.

[edit] Invariant construction of the Abel-Jacobi map

Let M be a manifold. Let π = π1(M) be its fundamental group. Let f: \pi \to \pi^{ab} be its abelianisation map. Let tor = tor(πab) be the torsion subgroup of πab. Let g: \pi^{ab} \to \pi^{ab}/tor be the quotient by torsion. Clearly,\pi^{ab}/tor= \mathbb{Z}^b, where b = b1(M). Let \phi=g \circ f : \pi \to \mathbb{Z}^b be the composed homomorphism.

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

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 integral harmonic 1-form along paths from a basepoint x_0\in M, we obtain a map to the circle \mathbb{R}/\mathbb{Z}=S^1.

Similarly, in order to define a map M\to H_1(M,\mathbb{R}) / H_1(M,\mathbb{Z})_{\mathbb{R}} without choosing a basis for cohomology, we argue as follows. Let x be a point in the universal cover \tilde{M} of M. Thus x is represented by a point of M together with a path c from x0 to it. By integrating along the path c, we obtain a linear form, h\to \int_c h, on E. We thus obtain a map \tilde{M}\to E^* = H_1(M,\mathbb{R}), which, furthermore, descends to a map

\overline{A}_M: \overline{M}\to E^*,\;\; c\mapsto \left( h\mapsto \int_c h \right),

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

Definition. The Jacobi variety (Jacobi torus) of M is the torus

J_1(M)=H_1(M,\mathbb{R})/H_1(M,\mathbb{Z})_\mathbb{R}.

Definition. The Abel-Jacobi map

A_M: M \to 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.

[edit] 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 ni are all positive integers, then

u(D) = u(E)\ if and only if D is linearly equivalent to E.

[edit] References

  • 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 978-0387909974.