Hopf curves and complex tori

Question

On Huybrecht's book there is an exercise asking to show that a Hopf curve $ X= \frac {\left ( \mathbb{C} \setminus \left \{ 0 \right \} \right )}{\mathbb{Z}}$ with the action, for $ k \in \mathbb{Z}$, $z \mapsto \lambda^{k}z$ with $\lambda > 0$ is isomorphic to a complex torus $ \frac{\mathbb{C}}{\Gamma}$ by determining $\Gamma$ explicitly. My manifold course didn't have discontinuous actions so I'm having some trouble with it, I was able to show that the Hopf curve is isomorphic to $\mathbb{S}^{1} \times \mathbb{S}^{1}$, but the isomprhism of it with the complex torus it is not done before on the book. I could show this, but I was wanting to calculate the lattice. I'm feeling that involves $\log_{\lambda}$, but I'm not geting anything bneyond this. Any hint would be appreciated.

Answer 1

This might be a little bit late, but I have been struggling with the same exercise and maybe it is overkill to open a new post. Let $X$ denote the Hopf curve as above. As Moishe suggested in the comments I have used an exponential map. Instead of just giving you the parameters I discovered I will expose the entire process: I propose $$f:\mathbb{C}\ni z \mapsto \exp(\alpha z)\in\mathbb{C}^*$$ for some $\alpha$ that I will fix later as I impose conditions. I want this map to cover a holomorphic homeomorphism $\mathbb{C}/\Lambda \rightarrow X$. For that I need $f$ to send equivalent elements in $\mathbb{C}/\Lambda$ to equivalent elements on $X=\mathbb{C}^*/\mathbb{Z}$ ($f$ needs to be equivariant).

I will abuse of some of the knowledge regarding complex structures on torii: we can assume without loss of generality that $\Lambda$ is spanned by $1,\tau$, with $\Im(\tau)>0$. I will write $\tau = \xi + i\zeta$. I now impose equivariance acknowledging that $w,w'\in \mathbb{C}^*$ are related if $w/w' \in \mathbb{R}^+$ and $\log(w/w')\in \lambda\mathbb{Z}$. So take $z$ and $z+1$, which are equivalent under the lattice relation. The condition $f(z)\sim f(z+1)$ is satisfied precisely if $\alpha$ is an integer multiple of $\lambda$, $$\alpha = m_\alpha \lambda.$$ Then I impose $f(z)\sim f(z+\tau)$. Again impose $f(z+\tau)/f(z)$ to be real and positive, and be proportional to some power of $\lambda$ to deduce that: $$ \xi = 2\pi\frac{m_\xi}{\alpha}$$ $$ \zeta = 2\pi\frac{m_\zeta}{\alpha}$$ Then we obtain that $f$ is equivariant under the corresponding equivalence relations. Since the complex structures on the quotients are induced by the standard complex structure on $\mathbb{C}$, it is fairly easy to see that $\frac{\partial}{\partial \overline{z}} f(z)=0$ and conclude $f$ is holomorphic. It descends to a bijective holomorphic map on the quotients which is therefore a biholomorphism.

DISCLAIMER: I have used three different parameters on the map and lattice to fit this together. Possibly this can be reduced to maybe two or even one parameters, but I am not so familiar with the space of complex structures on the torus so as to conclude something.