More Thoughts on Abel-Jacobi

This is an extension of another earlier post.  I think Jacobians of curves is a good way to introduce Abelian varieties.

Preliminaries: So $C$ is a complex projective integral variety of dimension 1 (a curve).  The sheafs of interest are $O = O_C, O^*, \mathscr{K}, \mathbb{Z}$ the structure sheaf, the sheaf of units, the (constant) sheaf of rational functions, and the constant sheaf $\mathbb{Z}$.

Set $g = \dim H^1(O) = \dim H^0(K)$, where $K$ is the canonical divisor for $C$.  Note, throughout I’ll truncate the notation $H^i(X,F) = H^i(F)$.  Also, I’ll tacitly use the natural isomorphism of $H^i(\mathbb{Z})$ with  singular cohomology of $C$ with coefficients in $\mathbb{Z}$.

Two short exact sequences of sheaves are of interest

$0 \to O \to \mathscr{K} \to \mathscr{K}/O \to 0$

$0 \to \mathbb{Z} \to O \xrightarrow{\exp} O^* \to 0$

Another standard fact, that wont be flushed out here is that $Pic C \cong H^1(O^*)$ and the natural cohomology map $H^1(O^*) \to H^2(\mathbb{Z}) \cong \mathbb{Z}$ gives the degree of the line bundle.

Degree zero line bundles.  It is of interest to look at degree zero line bundles, one good reason is they naturally form a subgroup $Pic^0 C \subset Pic C$.  Using the associated long exact sequence above, we see

$Pic^0 C = \ker \exp = H^1(O)/im(H^1(\mathbb{Z}))$

The short story is that $H^1(O) \cong \mathbb{C}^g$ and $H^1(\mathbb{Z}) \cong H_1(\mathbb{Z}) \cong \mathbb{Z}^{2g}$, so $Pic^0C$ is isomorphic to $\mathbb{C}^g/\Lambda$ for a lattice $\Lambda \cong \mathbb{Z}^{2g}$.

Describing the isomorphism $Pic^0C \cong \mathbb{C}^g/\Lambda$.  Using some nontrivial, but standard, results in algebraic geometry the above isomorphism can be described somewhat more explicitly.  Note what follows is more of an intuition and rough idea vs. something rigorous.  The results to use is that for a Noetherian seperated scheme, Cech cohomology can be used to compute $H^1(O)$ (since $O$ is quasicoherent).  Another result is Serre Duality, which in particular gives a perfect pairing $\alpha(, ) \colon H^1(O) \times H^o(K) \to \mathbb{C}$.

In order to describe $\alpha$, a slightly modified description of $H^0(\mathscr{K})$ and $H^0(\mathscr{K}/O)$ is used.  This is modified description comes from Rick Miranda and his book on complex algebraic curves.  A global section $f \in H^0(\mathscr{K}/O)$ is of course just a global meromorphic function.  So in local coordinates around any point $p \in C$, we can expand $f$ as a Laurent polynomial, $f(z_p) = \sum_i a_i z_p^i$.  Say we only keep track of $f(z_p)$ where there are some negative exponents, that is we keep the finitely many points where $f$ has a pole and throwaway the rest.  Now if $g$ is any other global meoromorphic function with same poles or the same order (i.e. same principle parts), then $g/f$ is a global holomophic function, so up to a constant we can recover $f$ from the Laurent Tail Divisor $\sum_{f(p) = \infty} f(z_p)\cdot p$

I’ll ignore this business of a constant, I don’t think it matter much.  In any case, applying the description to $H^0(\mathscr{K}/O)$, it follows that global sections can be expressed as $\sum_{i = 1}^n \bigl(\sum_{f_i(p) = \infty} f_i(z_p) \cdot p\bigr)$ where $f_i$ are local meromorphic functions defined on $U_i \subset C$, such that $f_i - f_j \in O(U_{ij})$ (the diff. is holomophic on the intersection).

Serre Duality. Recall elements of $H^1(O)$ are collection of elements $g_{ij} \in O(U_{ij})$ that agree on triple intersections.  Also, elements of $\eta \in H^0(K)$ are holomorphic 1-forms.  Also note, in the above description of global section of $H^0(\mathscr{K}/O)$, setting $g_{ij} = f_i - f_j$ gives an element of $H^1(O)$, and this is the map that appears in the long exact sequence in cohomology.  Now starting with $g_{ij} \in H^1(O)$, lift it to $H^0(\mathscr{K}/O)$ by fixing an index $j$, setting $f_j = 0$, and $f_i = g_{ij}$.  Finally we can say

$\alpha(g_{ij}, \eta) = \sum_{p \in C} \sum_{i, g_{ij}(p) = \infty} Res_p(g_{ij}\eta)$

The upshot is that summing over the residues of a function is like integrating it around a certain closed contour, and it can be checked (see post mentioned above) that the integral only depends on the class of the contour in $H_1(\mathbb{Z})$, so the pairing also looks like

$\alpha(g_{ij}, \eta) = \int_{\gamma} \eta$

So ultimately, if you fix a point $p \in C$ to be the base point of $\pi_1(C) \cong H_1(C)$, then you some of these maps become even more explicit (I guess) and the period lattice is determined by taking a basis for the differentials $\eta_1, ..., \eta_g$ and a generating set for $H_1(\mathbb{Z})$ $\gamma_1, ..., \gamma_{2g}$, then this is how you get the period matrix (or its transpose)

$P_{ij} = \int_{\gamma_i} \eta_j$.