---
tag: ZQ84M9D9TC
kind: concept
name: Jacobian variety
msc: [14H40, 14K30]
specializes: [R7W03TXRS9]
sources:
  - url: https://en.wikipedia.org/wiki/Jacobian_variety
same_as:
  wikidata: Q845729
  wikipedia: Jacobian variety
---

## Summary

The Jacobian variety of a compact [[Riemann surface|5K0N8QMV6Q]] of genus $g$ is a [[complex torus|Y7MA6KBE9M]] of dimension $g$.
It is built from the periods of the holomorphic differentials of the surface.
It is a principally polarized [[abelian variety|R7W03TXRS9]].
Its points correspond to divisor classes of degree $0$ on the curve, so it turns questions about divisors into questions about a group.
The [[Abel–Jacobi map|HE8F7KBF42]] maps the curve into its Jacobian.

## Explanations

### L1 human

Integrating the holomorphic differentials of a surface along loops gives a lattice of periods in $\C^g$.
Dividing $\C^g$ by this lattice gives a doughnut-shaped space of dimension $g$, which is the Jacobian.
It is a group, so points of the curve, and formal sums of them, can be added inside it.
For a torus $\C / \Lambda$ of genus $1$, the Jacobian is the torus itself.