---
tag: GB0D4XMZV3
kind: concept
name: Holomorphic fiber bundle
msc: [32L05, 55R10]
specializes: [CX4M8K61AB]
sources:
  - url: https://en.wikipedia.org/wiki/Fiber_bundle
---

## Summary

A holomorphic fiber bundle is a [[fiber bundle|CX4M8K61AB]] in the category of [[complex manifolds|JSGWAN2RJV]].
The total space, base and fibre are complex manifolds, and the local trivializations are biholomorphic.
[[Holomorphic vector bundles|TKA958Z4KZ]] 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 $\mathbb{P}^1$.

## Explanations

### L1 human

A holomorphic fiber bundle is a complex manifold that maps onto a base.
Seen from close up over the base, it looks like a product of a piece of the base with a fixed complex manifold, the fibre.
The fibres over different points are all the same complex manifold, but they may be twisted together globally.
For example, sending a nonzero vector of $\C^2$ to the line it spans has every fibre equal to a punctured line $\C^*$.
Globally this is not a product, although locally it is.