Abel–Jacobi map

The Abel–Jacobi map of a Riemannian manifold

Let ${\displaystyle M}$ be a smooth compact manifold. Let ${\displaystyle \pi ={\pi }_1(M)}$ be its fundamental group. Let ${\displaystyle f:\pi \to {\pi }^{ab}}$ be its abelianisation map. Let ${\displaystyle \mathrm{tor}=\mathrm{tor}({\pi }^{ab})}$ be the torsion subgroup of ${\displaystyle {\pi }^{ab}}$. Let ${\displaystyle g:{\pi }^{ab}\to {\pi }^{ab}/\mathrm{tor}}$ be the quotient by torsion. If ${\displaystyle M}$ is a surface, ${\displaystyle {\pi }^{ab}/\mathrm{tor}}$ is non-canonically isomorphic to ${\displaystyle {\mathbb{Z}}^{2g}}$, where ${\displaystyle g}$ is the genus; more generally, ${\displaystyle {\pi }^{ab}/\mathrm{tor}}$ is non-canonically isomorphic to ${\displaystyle {\mathbb{Z}}^b}$, where ${\displaystyle b}$ is the first Betti number. Let ${\displaystyle \phi =g\circ f:\pi \to {\mathbb{Z}}^b}$ be the composite homomorphism.

Definition. The cover ${\displaystyle \overline{M}}$ of the manifold ${\displaystyle M}$ corresponding to the subgroup ${\displaystyle \ker (\phi )\subset \pi }$ is called the universal (or maximal) free abelian cover.

Now assume ${\displaystyle M}$ has a Riemannian metric. Let ${\displaystyle E}$ be the space of harmonic 1-forms on ${\displaystyle M}$, with dual ${\displaystyle E^{\ast }}$ canonically identified with ${\displaystyle H_1(M,\mathbb{R})}$. By integrating an integral harmonic 1-form along paths from a basepoint ${\displaystyle x_0\in M}$, we obtain a map to the circle ${\displaystyle \mathbb{R}/\mathbb{Z}=S^1}$.

Similarly, in order to define a map ${\displaystyle M\to H_1(M,\mathbb{R})/H_1(M,\mathbb{Z}{)}_{\mathbb{R}}}$ without choosing a basis for cohomology, we argue as follows. Let ${\displaystyle x}$ be a point in the universal cover ${\displaystyle \tilde{M}}$ of ${\displaystyle M}$. Thus ${\displaystyle x}$ is represented by a point of ${\displaystyle M}$ together with a path ${\displaystyle c}$ from ${\displaystyle x_0}$ to it. By integrating along the path ${\displaystyle c}$, we obtain a linear form on ${\displaystyle E}$:

${\displaystyle h\to {\int }_{c}h.}$

This gives rise a map

${\displaystyle \tilde{M}\to E^{\ast }=H_1(M,\mathbb{R}),}$

which, furthermore, descends to a map

${\displaystyle \{\begin{array}{l}{\overline{A}}_M:\overline{M}\to E^{\ast }\\ c\mapsto \left(h\mapsto {\int }_{c}h\right)\end{array}}$

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

Definition. The Jacobi variety (Jacobi torus) of ${\displaystyle M}$ is the torus

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

Definition. The Abel–Jacobi map

${\displaystyle A_M:M\to J_1(M),}$

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. 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

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

Let ${\displaystyle M}$ denote a compact Riemann surface of genus ${\displaystyle g>0}$. Let ${\displaystyle \{a_1,...,a_g,b_1,...,b_g\}}$ be a canonical homology basis on ${\displaystyle M}$, and ${\displaystyle \{{\zeta }_1,...,{\zeta }_g\}}$ the dual basis for ${\displaystyle {\mathcal{H}}^1(M)}$, which is a ${\displaystyle g}$-dimensional complex vector space consisting of holomorphic differential forms. By a Dual basis we mean ${\displaystyle {\int }_{a_k}{\zeta }_j={\delta }_{jk}}$, for ${\displaystyle j,k=1,...,g}$. We can form a symmetric matrix whose entries are ${\displaystyle {\int }_{b_k}{\zeta }_j}$, for ${\displaystyle j,k=1,...,g}$. Let ${\displaystyle L}$ be the lattice generated by the ${\displaystyle 2g}$-columns of the ${\displaystyle g\times 2g}$ matrix whose entries consists of ${\displaystyle {\int }_{c_k}{\zeta }_j}$ for ${\displaystyle j,k=1,...,g}$ where ${\displaystyle c_k\in \{a_k,b_k\}}$. We call ${\displaystyle J(M)={\mathbb{C}}^g/L(M)}$ the Jacobian variety of ${\displaystyle M}$ which is a compact, commutative ${\displaystyle g}$-dimensional complex Lie group.

We can define a map ${\displaystyle \phi :M\to J(M)}$ by choosing a point ${\displaystyle P_0\in M}$ and setting ${\displaystyle \phi (P)=\left({\int }_{P_0}^P{\zeta }_1,...,{\int }_{P_0}^P{\zeta }_g\right).}$ 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 ${\displaystyle \mathrm{D}\mathrm{i}\mathrm{v}(M)}$ the divisor class group of ${\displaystyle M}$ then define a map ${\displaystyle \phi :\mathrm{D}\mathrm{i}\mathrm{v}(M)\to J(M)}$ by setting ${\displaystyle \phi (D)=\sum _{j=1}^r\phi (P_j)-\sum _{j=1}^s\phi (Q_j),D=P_1\cdots P_r/Q_1\cdots Q_s.}$

Note that if ${\displaystyle r=s}$ then this map is independent of the choice of the base point so we can define the base point independent map ${\displaystyle {\phi }_0:{\mathrm{D}\mathrm{i}\mathrm{v}}^{(0)}(M)\to J(M)}$ where ${\displaystyle {\mathrm{D}\mathrm{i}\mathrm{v}}^{(0)}(M)}$ denotes the divisors of degree zero of ${\displaystyle M}$.

The below Abel's theorem show that the kernel of the map ${\displaystyle {\phi }_0}$ is precisely the subgroup of principal divisors. Together with the Jacobi inversion problem, we can say that ${\displaystyle J(M)}$ is isomorphic as a group to the group of divisors of degree zero modulo its subgroup of principal divisors.

Abel–Jacobi theorem

The following theorem was proved by Abel (known as Abel's theorem): Suppose that

${\displaystyle D={\sum }_i{n}_i{p}_i}$

is a divisor (meaning a formal integer-linear combination of points of C). We can define

${\displaystyle u(D)={\sum }_i{n}_{i}u(p_i)}$

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 ${\displaystyle n_i}$ are all positive integers, then

${\displaystyle u(D)=u(E)}$ if and only if ${\displaystyle D}$ is linearly equivalent to ${\displaystyle E.}$ 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.