---
tag: 7RC9CN2Q49
kind: concept
name: Chern connection
msc: [53C07, 32L05, 53C55]
specializes: [5ZYRRVE8BW]
sources:
  - url: https://en.wikipedia.org/wiki/Hermitian_connection
---

## Summary

The Chern connection is a canonical [[connection|5ZYRRVE8BW]] on a holomorphic vector bundle with a Hermitian metric.
It is the unique connection that preserves the metric and whose $(0,1)$-part is the $\bar\partial$-operator of the bundle.
It turns the pair (holomorphic structure, metric) into curvature, a $(1,1)$-form with values in endomorphisms.
Its curvature is the basis of positivity notions, of vanishing theorems such as the [[Kodaira vanishing theorem|NY77DQ2AX8]], and of Chern–Weil representatives of [[Chern classes|PAD37GKDPD]].

## Explanations

### L1 human

On a bundle with a metric there are many ways to differentiate sections that keep lengths constant under parallel transport.
On a holomorphic bundle there is also a natural way to take the "antiholomorphic part" of a derivative.
The Chern connection is the only one that does both jobs at once.
For a line bundle with a local holomorphic section $e$ of squared length $H = \abs{e}^2$, it is given by $\nabla e = \partial \log H \otimes e$.
So the metric alone, written in a holomorphic frame, determines the connection and its curvature.