# Some thoughts on the the Abel-Jacobi map

Setup: C is a ‘nice’ curve over of genus g, and we’ve fixed a point p. For set be a path from p to q. vector space of holomorphic differential. Have

Not well defined. But well defined after modding out by periods: pick two different paths and , then the difference between the two answers we get is the functional

i.e. functional of form where c is a closed path; this only depends on the homology class of c.

pf: enough to show if where D is a triangulizable region, then for every w. Stokes theorem: . For w holomorphic, locally , so . QED

So there is a bijection . Anyways, the point of this post is to show we can can describe in a very canonical way….

Have:

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

.

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 . The map starts$

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

So modulo some details this canonical map is the same as the bijection from to cycles.

## About this entry

You’re currently reading “Some thoughts on the the Abel-Jacobi map,” an entry on Math Meandering

- Published:
- May 18, 2009 / 12:54 am

- Category:
- Teleman

- Tags:

## 1 Comment

Jump to comment form | comment rss [?] | trackback uri [?]