---
tag: RQRJ5H4Y3E
kind: concept
name: Stable vector bundle
msc: [14H60, 14J60, 14D20]
specializes: [P4DZM975BV]
sources:
  - url: https://en.wikipedia.org/wiki/Stable_vector_bundle
same_as:
  wikidata: Q2325488
  wikipedia: Stable vector bundle
---

## Summary

A stable vector bundle is a [[vector bundle|P4DZM975BV]] whose proper nonzero subbundles all have smaller slope than the bundle itself.
The slope of a bundle is its degree divided by its rank.
Stability is the condition that makes vector bundles on a curve form a good [[moduli space|FJZRW9P0K3]].
It was introduced by David Mumford in the setting of [[geometric invariant theory|9EG3A21ET1]].
Over a compact Riemann surface, the Narasimhan–Seshadri theorem identifies stable bundles of degree $0$ with irreducible unitary representations of the fundamental group.

## Explanations

### L1 human

A vector bundle on a curve has two numbers: its rank $r$ and its degree $d$, which measures how twisted it is.
The slope $d/r$ is the twist per dimension.
The bundle is stable when no smaller piece inside it is relatively more twisted than the whole.
A direct sum $L_1 \oplus L_2$ of line bundles with $\deg L_1 > \deg L_2$ is not stable, because $L_1$ is a piece with larger slope.
Stable bundles have only scalar automorphisms, which is why they can be classified by a well-behaved space.