# The Abel-Jacobi map

In the next couple of posts, I will write about Riemann surfaces and their Jacobian, following Griffiths and Harris. To each compact Riemann surface of genus , we will associate a -dimensional complex torus, which is a complex manifold of the form for a discrete full-rank integral lattice in .

Recall that by Dolbeault’s theorem, on a compact complex manifold there is an isomorphism

.

So on a Riemann surface of genus , taking -cohomology class we find that . Hence the dimension is

because

so since .

Let us first consider a genus compact Riemann surface, i.e. a torus. Then is 1-dimensional, let be a generator. If and are two points on the torus, of course the integral

is not well-defined as a complex number, it depends on the path chosen from to on which we integrate. Instead of fixing a path to make sense of this integral, we can change the space on which it lives to account for this indeterminacy: if are two paths from to , their difference will be a closed loop representing a cycle in . The idea is to view the integral as a point in the quotient .

More precisely, for a compact Riemann surface of genus , we consider cycles which give a basis for . We will suppose that is a **canonical** basis, i.e. that and for , where means the intersection product. This just means that intersects once positively and intersects no other cycles, think of the canonical cycles on the g-holed torus. The first half of the basis is the **-cycles** and the **-cycles** are the others, . Let’s also fix a basis for . Then the **periods** are the vectors

.

The **period matrix** is the matrix having the periods as columns:

.

Note that the periods are -linearly independent so they generate a full-dimensional **lattice**

.

To see this, recall that since , the forms generate . So if we had a relation with , this would give

and for all

so since the pairing is non-degenerate, this would give a relation , which is impossible since give a basis for the homology group.

The **Jacobian** of the Riemann surface is defined to be the **complex torus**

.

Intrinsically, is just the quotient where we identify with its image under the natural inclusion which takes a class of closed loops and maps it to the functional . This space is the right one to consider integrals on in the sense that the vector

is not well-defined as a point of but it is if we consider is as living on , i.e. **modulo the period lattice** .

After choosing a base point , we can define a mapping

by

(mod ).

In fact, given a **divisor** , we can extend this map by letting . In particular, for divisors of degree 0

,

the map is independent of the base point and

(mod ).

When considered on divisors of degree 0, this is the **Abel-Jacobi map**

.

The first goal will be to show **Abel’s theorem**, which says that the kernel of is exactly the divisors of meromorphic functions on . Recall the short exact sequence

where is the space of all meromorphic functions on . If we restrict this to and the corresponding flat line bundles , Abel’s theorem gives an injection

.

Our second goal will be to show **Jacobi’s inversion theorem** which essentially says that this injection is in fact an isomorphism. From this point of view, we see that the -dimensional torus naturally parametrizes flat line bundles on . In fact, **Torelli’s theorem** says that , considered as a **principally polarized abelian variety**, entirely determines the Riemann surface we started with. Jacobian varieties are thus key in the study of compact Riemann surfaces.

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