---
tag: F5R676980R
kind: definition
name: Holomorphic line bundle
msc: [32L05, 14C22]
defines: QAC7BYK9KX
uses: [TKA958Z4KZ, JSGWAN2RJV, BDB06N91FN]
sources:
  - url: https://en.wikipedia.org/wiki/Holomorphic_vector_bundle
---

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]].
A *holomorphic line bundle* on $X$ is a [[holomorphic vector bundle|TKA958Z4KZ]] $\pi \colon L \to X$ of rank $1$.

## Remarks

Equivalently, $L$ is given by an open cover $(U_i)$ of $X$ and nowhere vanishing holomorphic functions $g_{ij} \colon U_i \cap U_j \to \C^*$ with $g_{ij} g_{jk} = g_{ik}$ on $U_i \cap U_j \cap U_k$.

Example: the [[tautological line bundle|BDB06N91FN]] $\mathcal{O}(-1)$ on $\mathbb{P}^n$ is $\set{(\ell, v) \in \mathbb{P}^n \times \C^{n+1} : v \in \ell}$ with the projection to $\mathbb{P}^n$.
On $U_i = \set{z_i \ne 0}$ the point $(\ell, v)$ goes to $(\ell, v_i)$.
The fibre coordinates satisfy $v_i = (z_i / z_j) v_j$, and $z_i / z_j$ is holomorphic and nonzero on $U_i \cap U_j$.
Its dual is $\mathcal{O}(1)$, and $\mathcal{O}(k)$ is the $k$-th tensor power of $\mathcal{O}(1)$.

Non-example: the holomorphic tangent bundle of $\mathbb{P}^2$ is a holomorphic vector bundle of rank $2$, so it is not a line bundle.
Its top exterior power is a line bundle, the anticanonical bundle $\mathcal{O}(3)$.

The tensor product of two line bundles with transition functions $g_{ij}$ and $h_{ij}$ has transition functions $g_{ij} h_{ij}$, and the dual of the first has $g_{ij}^{-1}$.