Given a holomorphic map $$\phi:\frac{\mathbb{C}}{\Lambda}\to\frac{\mathbb{C}}{\Lambda'}$$

My aim is to lift it to a holomorphic map $\Phi:\mathbb{C}\to\mathbb{C}$. The author has hinted to use the lifting property for this.

I know that if $\tilde{X}$ is a covering space of $X$ then any map $f:Y\to X$ can be lifted to a map $\tilde{f}:Y\to\tilde{X}$. Since $\mathbb{C}$ is the covering space of any complex tori with covering map being the canonical projection map (correct me if I'm wrong). Hence my aim is to find a holomorphic $\Phi$ such that the diagram commutes.

$$\begin{array} A\mathbb{C} & \stackrel{\Phi}{\longrightarrow} & \mathbb{C} \\ \downarrow{p_1} & & \downarrow{p_2} \\ \frac{\mathbb{C}}{\Lambda} & \stackrel{\phi}{\longrightarrow} & \frac{\mathbb{C}}{\Lambda'} \end{array} $$ How do I proceed? Thank you.

Edit: I recently checked this question. How can we show that the lift of this holomorphic function will be holomorphic?


The question you linked gives us the existence of the map you want. For it to be holomorphic it is enough to see that $p_1$ and $p_2$ are holomorphic and locally biholomorphic.

  • $\begingroup$ But if $p_1$ is holomorphic and doubly periodic, it will turn out to be constant, right? $\endgroup$ – Swapnil Tripathi Nov 18 '16 at 14:13
  • $\begingroup$ You are defining the complex structure on the tori using the functions $p_1$ and $p_2$. They are just coordinates on the manifolds and the function $\phi$ is going to be holomorphic with respect to the coordinates $p_1$ and $p_2$. No $p_1$ and $p_2$ are not constants, liuville's theorem doesn't apply here. $\endgroup$ – hjr Nov 18 '16 at 14:21
  • $\begingroup$ I am having a major crisis in my head (I tried visualizing the map which seems wrong now). How do I proceed to show that $p_1$ is holomorphic? Maybe that'll give me some ideas. $\endgroup$ – Swapnil Tripathi Nov 18 '16 at 22:35
  • $\begingroup$ So what you need to show that $p_1$ is holomorphic is that for any point $z\in\mathbb{C}$ then there are open sets $U,V$. $U$ a neighborhood of $z$, $V$ a neighborhood of $p_1(z)$ and a chart $\psi:V\to W$ ($W$ a subset of $\C$ and $\psi$ a homeomorphism). Then $p_1$ is holomorphic at $z$ if $\psi\circ\p_1|U:U\to W$ is holomorphic. $\endgroup$ – hjr Nov 19 '16 at 5:17
  • $\begingroup$ Pressed enter too quickly, dont know how to edit my comment. But anyway, $\endgroup$ – hjr Nov 19 '16 at 5:18

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