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# Stable vector bundle

In mathematics, a stable vector bundle is a vector bundle that is stable in the sense of geometric invariant theory. They were defined by

## Stable vector bundles over curves

A bundle W over an algebraic curve (or over a Riemann surface) is stable if and only if

$displaystylefrac\left\{deg\left(V\right)\right\}\left\{hbox\left\{rank\right\}\left(V\right)\right\} < frac\left\{deg\left(W\right)\right\}\left\{hbox\left\{rank\right\}\left(W\right)\right\}$

for all proper non-zero subbundles V of W and is semistable if

$displaystylefrac\left\{deg\left(V\right)\right\}\left\{hbox\left\{rank\right\}\left(V\right)\right\} le frac\left\{deg\left(W\right)\right\}\left\{hbox\left\{rank\right\}\left(W\right)\right\}$

for all proper non-zero subbundles V of W. Informally this says that a bundle is stable if it is "more ample" than any proper subbundle, and is unstable if it contains a "more ample" subbundle. The moduli space of stable bundles of given rank and degree is an algebraic variety. showed that stable bundles on projective nonsingular curves are the same as those that have projectively flat unitary irreducible connections; these correspond to irreducible unitary representations of the fundamental group. Kobayashi and Hitchin conjectured an analogue of this in higher dimensions; this was proved for projective nonsingular surfaces by , who showed that in this case a vector bundle is stable if and only if it has an irreducible Hermitian-Einstein connection.

The cohomology of the moduli space of stable vector bundles over a curve was described by and .

## Stable vector bundles over projective varieties

If X is a smooth projective variety of dimension n and H is a hyperplane section, then a vector bundle (or torsionfree sheaf) W is called stable if

$frac\left\{chi\left(V\left(nH\right)\right)\right\}\left\{hbox\left\{rank\right\}\left(V\right)\right\} < frac\left\{chi\left(W\left(nH\right)\right)\right\}\left\{hbox\left\{rank\right\}\left(W\right)\right\}text\left\{ for \right\}ntext\left\{ large\right\}$

for all proper non-zero subbundles (or subsheaves) V of W, and is semistable if the above holds with < replaced by ≤.

## References

• especially appendix 5C.
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