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Suppose $B\subset\mathbb{C}$ is a contractible open set, and there is a complex manifold $X$ and a holomorphic map $\pi: X \to B$ such that each fiber $\pi^{-1}\{b\}$ isomorphic to $\mathbb{C}$, i.e. $\pi$ is a holomorphic fiber bundle. My questions are
Does there alway exist a holomorphic secti...