I want to show the quotient $L:=\left(\mathbb{R}^2 \times \mathbb{C}\right) / \mathbb{Z}^2$ is a complex line bundle over $\mathbb{T}^2$. With the group action given by: $$ (m, n) \cdot(x, y, z)=\left(x+m, y+n, e^{2 \pi i n x} z\right), \quad(m, n) \in \mathbb{Z}^2,(x, y, z) \in \mathbb{R}^2 \times \mathbb{C} $$ I want to find local frames and show that the transition function between local frame is holomorphic. My guess is that the local frame is something like this: $$e_U ([x,y]) = e^{2\pi i x}$$
Any suggestions (hint) on which chart and local frame should I construct here?