Let $\mathbb{C} \backslash \mathbb{Z}^2$ be the usual torus, seen as a holomorphic manifold. What are the most usual local holomorphic charts that are used to study this holomorphic manifold ?
I have the same question for the general torus $\mathbb{C}^n \backslash \Lambda$. I've seen a lot of documentation about these torus but not the charts written in an explicit way.
Choose a $\mathbb Z$-basis $\lambda_1,\cdots ,\lambda_{2n}$ of $\Lambda$ and consider the projection $\pi:\mathbb C^n\to X=\mathbb C^n/\Lambda$.
For each $a\in\mathbb C^n$ define the open subset $V_a=\{a+\sum_{j=1}^{2n}r_j\lambda_j\vert 0\lt r_j\lt1 \}\subset \mathbb C^n$ and put $U_a=\pi(V_a)\subset X,\: \pi_a=(\pi\vert V_a): V_a\stackrel {\sim}{\to} U_a$.
The homeomorphisms $\pi_a^{-1}: U_a\to V_a \;({a\in \mathbb C^n})$ are the required canonical holomorphic atlas for the torus $X$.