---
tag: FVCDGMNSAN
kind: concept
name: Griffiths positivity
msc: [32L05, 53C55]
sources:
  - url: https://en.wikipedia.org/wiki/Ample_line_bundle
---

## Summary

Griffiths positivity is a notion of positive curvature for a holomorphic Hermitian vector bundle of any rank.
It asks that the curvature of the [[Chern connection|7RC9CN2Q49]] be positive when evaluated on one tangent vector and one fibre vector at a time.
For line bundles it is the same as being a [[positive line bundle|J45KWFHENP]].
Griffiths positive bundles are ample, and Griffiths asked whether the converse holds; this question is open in general.

## Explanations

### L1 human

For a line bundle, curvature is a single $2$-form, and positivity has one clear meaning.
For a bundle of higher rank, curvature is a matrix of $2$-forms, and there are several reasonable ways to call it positive.
Griffiths positivity is the weakest common one: for each direction on the base and each vector in the fibre, a certain number built from the curvature must be positive.
A direct sum of copies of a positive line bundle, with the sum metric, is Griffiths positive.