---
tag: X2E0QP770D
kind: definition
name: Complex torus
msc: [14K20, 32J18]
defines: Y7MA6KBE9M
uses: [WN9V7WSJKF, 0KYX2N69J0]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_torus
---

## Definition

Let $g \ge 1$ be an integer.
A *lattice* in $\C^g$ is a subgroup $\Lambda \subseteq \C^g$ generated by $2g$ vectors that are linearly independent over $\R$.
A *complex torus* of dimension $g$ is the [[quotient group|0KYX2N69J0]] $X = \C^g / \Lambda$ for a [[lattice|WN9V7WSJKF]] $\Lambda$ in $\C^g$.
It carries the complex manifold structure for which the projection $\C^g \to X$ is a local biholomorphism.

## Remarks

Example: $\Lambda = \Z + \Z \tau$ with $\tau \in \C$ and $\operatorname{Im} \tau > 0$.
The numbers $1$ and $\tau$ are linearly independent over $\R$, so $\Lambda$ is a lattice in $\C$, and $\C / \Lambda$ is a complex torus of dimension $1$.
Every complex torus of dimension $1$ is the set of complex points of an elliptic curve.

Non-example: $\Z \subseteq \C$ is a discrete subgroup, but it is generated by only one vector, not two.
The quotient $\C / \Z$ is not compact; the map $z \mapsto e^{2 \pi i z}$ identifies it with $\C^{*}$.

Let $X = \C^g / \Lambda$ be a complex torus.
Then $X$ comes from an [[abelian variety|R7W03TXRS9]] if and only if $\C^g$ has a positive definite Hermitian form $H$ with $\operatorname{Im} H(\Lambda \times \Lambda) \subseteq \Z$.
Such a form is called a Riemann form or a polarization.