---
tag: Y7MA6KBE9M
kind: concept
name: Complex torus
msc: [14K20, 32J18]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_torus
same_as:
  wikidata: Q5156610
  wikipedia: Complex torus
---

## Summary

A complex torus of dimension $g$ is the quotient of $\C^g$ by a [[lattice|WN9V7WSJKF]] of rank $2g$.
It is a compact connected complex manifold and a commutative complex Lie group.
Every complex [[abelian variety|R7W03TXRS9]] is a complex torus, but for $g \ge 2$ most complex tori are not algebraic.

## Explanations

### L1 human

Take the plane $\C$ and the grid of points $m + n i$ with integers $m$ and $n$.
Glue together any two points that differ by a grid point.
Each square of the grid is folded up into the surface of a doughnut, the quotient $\C / (\Z + \Z i)$.
A complex torus does the same in $\C^g$, with a grid that has $2g$ independent directions.
Addition in $\C^g$ passes to the quotient, so the points of a complex torus form a group.