Stable vector bundle

In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable bundles were defined by David Mumford in Mumford (1963) and later built upon by David Gieseker, Fedor Bogomolov, Thomas Bridgeland and many others.

Motivation

On a smooth projective variety, line bundles of given numerical invariants are parametrised over a well-behaved moduli space (isomorphic to the Picard variety). This space is, in particular, a proper and separated scheme of finite type, thus lending itself to algebro-geometric analysis. Similar considerations fail when naively parametrising vector bundles of higher rank.

As an example, consider the moduli of vector bundles of rank

r

=

2

{\displaystyle r=2}

and first Chern class

c

1

=

0

{\displaystyle c_{1}=0}

on the complex projective line

P

1

{\displaystyle \mathbb {P} ^{1}}

. If they were to form a separated moduli space, the valuative criterion would imply that any family over the punctured line

C

∗

{\displaystyle \mathbb {C} ^{\ast }}

can be completed to at most one family over

C

{\displaystyle \mathbb {C} }

. But it is straightforward to construct a family that admits two non-isomorphic completions.

Indeed consider the constant family assigning each

t

∈

C

∗

{\displaystyle t\in \mathbb {C} ^{\ast }}

to the bundle

V

t

≃

O

⊕

O

{\displaystyle V_{t}\simeq {\mathcal {O}}\oplus {\mathcal {O}}}

.