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The Abel-Jacobi map

May 5, 2014

In the next couple of posts, I will write about Riemann surfaces and their Jacobian, following Griffiths and Harris. To each compact Riemann surface S of genus g \geq 1, we will associate a g-dimensional complex torus, which is a complex manifold of the form \mathbb{C}^g/\Lambda for \Lambda a discrete full-rank integral lattice in \mathbb{C}^g.

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

H^q(M,\Omega^p) \cong H_{\bar{\partial}}^{p,q}(M).

So on a Riemann surface S of genus g \geq 1, taking \bar{\partial}-cohomology class we find that H^0(S,\Omega^1) \cong H^{1,0}_{\bar{\partial}}(S). Hence the dimension is

\dim H^0(M,\Omega^1) = g

because

H^1(S,\mathbb{C}) \cong H^{1,0}(S) \oplus H^{0,1}(S)

so 2h^{1,0}(S) = h^1(S) = 2 - \chi(S) = 2g since \overline{H^{1,0}(S)} = H^{0,1}(S).

Let us first consider C a genus 1 compact Riemann surface, i.e. a torus. Then H^0(S,\Omega^1) is 1-dimensional, let \omega be a generator. If p and q are two points on the torus, of course the integral

\displaystyle\int_q^p \omega

is not well-defined as a complex number, it depends on the path chosen from q to p 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 \gamma_1,\gamma_2 are two paths from q to p, their difference will be a closed loop representing a cycle in H_1(C,\mathbb{Z}). The idea is to view the integral as a point in the quotient \mathbb{C}/H_1(C,\mathbb{Z}).

More precisely, for a compact Riemann surface S of genus g, we consider cycles \delta_1, \ldots, \delta_{2g} which give a basis for H_1(S,\mathbb{Z}). We will suppose that (\delta_i) is a canonical basis, i.e. that [\delta_i] \cdot [\delta_{i+g}] = 1 and [\delta_i] \cdot [\delta_j] = 0 for j \neq i+g, where \cdot means the intersection product. This just means that \delta_i intersects \delta_{i+g} once positively and intersects no other cycles, think of the canonical cycles on the g-holed torus. The first half of the basis \delta_1, \ldots, \delta_{g} is the A-cycles and the B-cycles are the others, \delta_{g+1}, \ldots, \delta_{2g}. Let’s also fix a basis \omega_1, \ldots, \omega_g for H^0(S,\Omega^1). Then the periods are the vectors

\Pi_i = \left( \displaystyle\int_{\delta_i}\omega_1, \ldots, \displaystyle\int_{\delta_i}\omega_g \right) \in \mathbb{C}^g.

The period matrix is the g \times 2g matrix \Omega having the periods as columns:

\Omega = (\Pi_1 \cdots \Pi_{2g}).

Note that the periods are \mathbb{R}-linearly independent so they generate a full-dimensional lattice

\Lambda = \{m_1\Pi_1 + \cdots + m_{2g}\Pi_{2g} \mid m_i \in \mathbb{Z} \}.

To see this, recall that since H^1(S,\mathbb{C}) = H^{1,0}(S) \oplus H^{0,1}(S), the forms \{\omega_i, \overline{\omega}_i\} generate H^1(S,\mathbb{C}). So if we had a relation \sum_ik_i\Pi_i = 0 with k_i \in \mathbb{R}, this would give

\displaystyle\sum_i k_i \displaystyle \int_{\delta_i} \omega_j = 0 and \displaystyle\sum_i k_i \displaystyle\int_{\delta_i} \overline{\omega}_i = 0 for all j

so since the pairing (\omega, \delta) \mapsto \int_{\delta} \omega is non-degenerate, this would give a relation \sum_i k_i [\delta_i] = 0 \in H_1(S,\mathbb{R}), which is impossible since (\delta_i) give a basis for the homology group.

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

J(S) = \mathbb{C}^g/\Lambda.

Intrinsically, J(S) is just the quotient (H^0(S,\Omega^1)^*/H_1(S,\mathbb{Z}) where we identify H_1(S,\mathbb{Z}) with its image under the natural inclusion which takes a class of closed loops [\gamma] and maps it to the functional \theta \mapsto \int_{\gamma} \theta. This space is the right one to consider integrals on S in the sense that the vector

\left( \displaystyle\int_q^p \omega_1, \ldots, \displaystyle\int_q^p \omega_g \right)

is not well-defined as a point of \mathbb{C}^g but it is if we consider is as living on J(S), i.e. modulo the period lattice \Lambda.

After choosing a base point p_0 \in S, we can define a mapping

\mu : S \to J(S)

by

\mu(p) = \left( \displaystyle\int_{p_0}^p \omega_1, \ldots, \displaystyle\int_{p_0}^p \omega_g \right)    (mod \Lambda).

In fact, given a divisor D = \sum_i a_ip_i \in \text{Div}(S), we can extend this map by letting \mu(D) = \sum_i a_i \mu(p_i). In particular, for divisors of degree 0

D = \displaystyle\sum_i (p_i - q_i) \in \text{Div}_0(S),

the map \mu is independent of the base point and

\mu(D) = \displaystyle\sum_i \left( \displaystyle\int_{q_i}^{p_i} \omega_1, \ldots, \displaystyle\int_{q_i}^{p_i} \omega_g \right)   (mod \Lambda).

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

\mu : \text{Div}_0(M) \to J(S).

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

\mathcal{M}(S) \longrightarrow \text{Div}(S) \longrightarrow \text{Pic}(S) \to 0

where \mathcal{M}(S) is the space of all meromorphic functions on S. If we restrict this to \text{Div}_0(S) and the corresponding flat line bundles \text{Pic}_0(S), Abel’s theorem gives an injection

\text{Pic}_0(S) \cong \text{Div}_0(S)/\mathcal{M}(S) \hookrightarrow J(S).

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 g-dimensional torus J(S) naturally parametrizes flat line bundles on S. In fact, Torelli’s theorem says that J(S), considered as a principally polarized abelian variety, entirely determines the Riemann surface S we started with. Jacobian varieties are thus key in the study of compact Riemann surfaces.

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