# Some thoughts on the the Abel-Jacobi map

Setup: C is a ‘nice’ curve over $\ \mathbb{C}$ of genus g, and we’ve fixed a point p.  For $\ q \in C$ set  $\ \gamma_{pq}$ be a path from p to q.  $\ \Omega_C$ vector space of holomorphic differential.  Have

$\ J \colon C \to \Omega_C^*$

$\ q \mapsto \int_{\gamma_{pq}}$

Not well defined.  But well defined after modding out by periods: pick two different paths $\ p \xrightarrow{f} q$ and $\ p \xrightarrow{g} q$, then the difference between the two answers we get is the functional

$\ w \mapsto \int_{p \xrightarrow{f} q \xrightarrow{g^{-1}} p } w$

i.e. functional of form $\ \int_c (-)$ where c is a closed path; this only depends on the homology class of c.

pf: enough to show if $\ c = \partial D$ where D is a triangulizable region, then $\ \int_{c}w = 0$ for every w.  Stokes theorem: $\ \int_c w = \int_D dw$.  For w holomorphic, locally $\ w = f(z)dz$, so $\ dw = (\partial + \overline{\partial})w = \frac{\partial f}{\partial \overline{z}}d\overline{z}\wedge dz = 0$. QED

So there is a bijection $\ H_1(C, \mathbb{Z}) \to \Gamma$.  Anyways, the point of this post is to show we can can describe$\ H_1(C,\mathbb{Z}) \to \Gamma$ in a very canonical way….

Have:

$\ H_1(C,\mathbb{Z}) \to \hom(H_1, \mathbb{Z}) \cong H^1(C, \mathbb{Z}) \to H^1(C, \mathbb{C})$

The first map is given by intersection pairing: (a |–> #(?. a) ).  The last map is a natural inclusion.  Also

$\ H^1(C, \mathbb{C}) \leftarrow H^1_{DR}(C) = \Omega_C^* \oplus \overline{\Omega}_C^*$.

The first map is an isomorphism and is the content of the de Rham Theorem (pg. 43 Griff & Harr).  The last equality is the Dolbeault Theorem ( pg. 45 Griff & Harr) plus Hodge decomposition (pg. 116 Griff & Harr).  We can now follow these maps and see if its the same as $\ c \mapsto \int_c$.  The map starts\$

$\ c \mapsto \#(?.c) = \int_? \phi \in H^1(C, \mathbb{C})$

Where $\ \phi$ is the Poincare Dual  (again, see Griff & Harr) to c.  Which is to say $\ \#(b.c) = \int_b \phi$ for every b.  But by a Kunneth Formula, $\ \#(b.c) = \int_C \psi\wedge \phi$, where $\ \psi$ is Poicare dual to b.  Putting these last two things together shows $\ \int_b (?) = \int_C \psi \wedge ?$.  So applying this to c, we get our map is

$\ c \mapsto \#(?.c) = \int_C ? \wedge \phi = \int_c$

So modulo some details this canonical map is the same as the bijection from $\ H_1(C)$ to cycles.