I saw that the (primary) Hopf surface is an elliptic surface over $\mathbb{P}^1$. I want to see why. What I am thinking is by the Hopf fibration $S^1\to S^3\to S^2$, we can construct a fibration $S^1\times S^1\to S^3\times S^1\to S^2$ by taking the product of each fiber and $S^1$. And $S^3\times S^1$ is the smooth structure of the Hopf surface, $S^2$ is the smooth structure of $\mathbb{P}^1$, $S^1\times S^1$ is the smooth structure of the elliptic curve. I wonder if we can show this bundle map is holomorphic so that the fibration $S^1\times S^1\to S^3\times S^1\to S^2$ gives the claim that Hopf surface is an elliptic surface over $\mathbb{P}^1$?
Do I have to check the local formula to determine if the map $S^3\times S^1$ is holomorphic or is there an easier way to see it?
The primary Hopf surface I mean is the compact complex surface given by $\mathbb C^2-\{0\}/\mathbb Z$ where $\gamma:(z_1,z_2)\mapsto (2z_1,2z_2)$ generates $\mathbb Z$. And it can be proved that it's diffeomorphic to $S^3\times S^1$.
Attempt: I am thinking maybe I can use the quotient manifold theorem. See the fiber as a complex Lie group, $T:=\mathbb C/(\mathbb Z+\tau\mathbb Z)$ and it acts on $\mathbb C^2-\{0\}/\mathbb Z$ properly and freely, then we get a fiber bundle $\mathbb C/(\mathbb Z+\tau\mathbb Z)\to \mathbb C^2-\{0\}/\mathbb Z\to (\mathbb C^2-\{0\}/\mathbb Z)/T$ and by the long exact sequence of homotopy groups we can see $(\mathbb C^2-\{0\}/\mathbb Z)/T$ is $\mathbb{P}^1$. And it's easy to see the action is free, so now the question is if the action is proper? And if so, then by the quotient manifold theorem, we get a holomorphic fiber bundle $T\to \mathbb C^2-\{0\}/\mathbb Z\to\mathbb P^1$