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.

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