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.