Holomorphic fiber bundle

Summary

A holomorphic fiber bundle is a fiber bundle in the category of complex manifolds. The total space, base and fibre are complex manifolds, and the local trivializations are biholomorphic. Holomorphic vector bundles are the case where the fibre is $\C^r$ and the transition maps are linear. Other examples are the Hopf map $\C^2 \setminus \set{0} \to \mathbb{P}^1$ and the elliptic fibration of a Hopf surface over $\mathbb{P}^1$.