I have a question about a complex manifold $M$ and a holomorphic submersion $p : M \to S$ to a compact complex curve satisfying the following conditions:

- $p^{-1}(x)$ is biholomorphic to $\mathbb{C}$ for any $x \in S$.
- For any $z \in M$, there is a holomorphic section $s(z)$, i.e., a compact holomorphic submanifold of $M$ such that $p : s(z) \to S$ is a holomorphic equivalence.
- $M$ is a smooth $\mathbb{R}^2$ bundle whose fiber is given by $p^{-1}(x)$, $x \in S$.

Can we deduce that $M$ is biholomorphic to $S \times \mathbb{C}$?

Remark: Inspired by Speyer's insightful example, where $M$ is a non-trivial line bundle, I am now introducing additional conditions and asking whether $M$ is biholomorphic to $S \times \mathbb{C}$:

- The global section $s(z)$ is nowhere vanishing for any $z\in M$.
- $M$ is a $C^\infty$ trivial $\mathbb{R}^2$ bundle over $S$.