---
tag: J45KWFHENP
kind: concept
name: Positive line bundle
msc: [32L05, 32Q15, 14C20]
specializes: [QAC7BYK9KX]
sources:
  - url: https://en.wikipedia.org/wiki/Ample_line_bundle
---

## Summary

A holomorphic line bundle is positive when it has a Hermitian metric whose [[Chern connection|7RC9CN2Q49]] has positive curvature.
Then $i$ times the curvature is a Kähler form, so a compact complex manifold with a positive line bundle is a [[Kähler manifold|X60JG9AEQJ]].
By the [[Kodaira embedding theorem|7G7CSM2NZ9]], on a compact complex manifold the positive line bundles are exactly the [[ample line bundles|RRK8P6TXG2]].
Positivity is the analytic hypothesis of the Kodaira and Kodaira–Nakano vanishing theorems.

## Explanations

### L1 human

The curvature of a line bundle with a metric is a $2$-form, and "positive" means that this form is like a Kähler form at every point.
The model is the line bundle $\mathcal{O}(1)$ on projective space, whose curvature is the Fubini–Study form.
Positive bundles have many holomorphic sections in their high powers, enough to embed the manifold in projective space.
The trivial bundle on a compact manifold is never positive: its curvature can be made zero, but never strictly positive everywhere.