I came across the following description of a (1-dimensional) complex torus while learning about Calabi-Eckmann manifolds:
For a fixed $\alpha \in \mathbb{C} \setminus \mathbb{R}$, the subgroup $Z = \{(e^t , e^{\alpha t}) \mid t \in \mathbb{C} \} \subseteq \mathbb{C}^* \times \mathbb{C}^*$ is complemented by $S^1 \times S^1$ so that $M = \mathbb{C}^* \times \mathbb{C}^* / Z$ is isomorphic to the torus $S^1 \times S^1$.
In a Google search, I found an off-hand remark that for the same $\alpha \in \mathbb{C} \setminus \mathbb{R}$, $M$ is isomorphic to the usual complex torus $\mathbb{C} / \{ n + m \alpha \mid n , m \in \mathbb{Z} \}$.
I haven't been able to figure out an obvious isomorphism between these two groups that preserves their complex structures. Can anyone help me out?
Let $f : \Bbb C \to \Bbb C^{*2}$ be defined by $f(z) = (1,\exp(2i\pi z))$
It is easy to check that $f(z+1) = f(z)$, and $f(z+a) = f(z)(1,\exp(2i\pi\alpha)) \in f(z)Z$, and so $f$ induces a morphism $\tilde f : \Bbb C/\langle 1,\alpha \rangle \to M$
$\tilde f$ is surjective : if $(\exp u, \exp v) \in \Bbb C^{*2}$ then we have $(\exp u, \exp v) = (\exp u, \exp (\alpha u)) (1, \exp(v-\alpha u))$. The first term is in $Z$ and the second is $f((v-\alpha u)/2i\pi)$
$\tilde f$ is injective : if $f(z_1)f(z_2)^{-1} \in Z$ then $(1,\exp(2i\pi(z_1-z_2))) \in Z$ so there is an integer $k$ such that $\exp(2i\pi(z_1-z_2)) = \exp(2ik\pi\alpha)$, and in turn there is an integer $l$ such that $2i\pi(z_1-z_2) = 2i\pi(k\alpha + l)$, and so $z_1-z_2 \in \langle 1,\alpha \rangle$