---
tag: TKA958Z4KZ
kind: concept
name: Holomorphic vector bundle
msc: [32L05, 32L10]
specializes: [P4DZM975BV]
sources:
  - url: https://en.wikipedia.org/wiki/Holomorphic_vector_bundle
same_as:
  wikidata: Q9289328
  wikipedia: Holomorphic vector bundle
---

## Summary

A holomorphic vector bundle is a complex [[vector bundle|P4DZM975BV]] over a [[complex manifold|JSGWAN2RJV]].
Its total space is a complex manifold, and its local trivializations are holomorphic.
Equivalently, its transition functions are holomorphic maps into $GL_r(\C)$.
Holomorphic vector bundles carry sheaves of holomorphic sections, and their cohomology is a main tool of complex geometry.
Examples are the holomorphic tangent bundle, the bundles of holomorphic forms, and the line bundles $\mathcal{O}(k)$ on projective space.

## Explanations

### L1 human

Over each point of a complex manifold sits a complex vector space, and these spaces vary holomorphically from point to point.
Locally the bundle looks like a product $U \times \C^r$, and two such local pictures are glued by matrices whose entries are holomorphic functions.
The holomorphic sections are then the "vector-valued holomorphic functions" of the bundle.
On the Riemann sphere, the tautological line bundle, whose fibre over a line in $\C^2$ is that line, is the simplest nontrivial example.