Complex torus

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 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.