bundles

complex geometry

# Contents

## Idea

A vector bundle (typically considered in complex-analytic geometry or algebraic geometry) is called (semi-)stable if it is a (semi-)stable point in the moduli space of bundles. Under suitable conditions this is equivalent to a certain inequality on the slopes of the sub-bundles, and this inequality is what tends to be stated as the definition of stability of vector bundles.

## Definition

### Over a Riemann surface / over an algebraic curve

For $\Sigma$ a Riemann surface, a complex vector bundle $E \to \Sigma$ over $\Sigma$ is called (slope-)stable if for all non-trivial subbundles $K \hookrightarrow E$ the inequality

$\mu(K) \lt \mu(E)$

between their slopes holds, i.e. if the inequality

$\frac{deg(K)}{rank(K)} \lt \frac{deg(E)}{rank(E)}$

holds between the fractions of degree and rank of the vector bundles holds.

## Examples

Every line bundle is slope-stable. The extension of a degree-0 line bundle by a degree-1 line bundle is stable.

## Properties

### Relation to GIT-stability

The slope-(semi-)stable vector bundles are essentially the (semi-)stable points in the sense of geometric invariant theory in the moduli space of bundles. The precise statement is reviewed for instance in (Saiz 09, section 2.3.

### Relation to connections

The Narasimhan–Seshadri theorem identifies moduli spaces of stable vector bundles over complex curves with those of certain flat connections.

The Donaldson-Uhlenbeck-Yau theorem relates semi-stable vector bundles over Kähler manifolds to Hermite-Einstein connections.

Still more generally, the Kobayashi-Hitchin correspondence relates semi-stable vector bundles over complex manifolds to Hermite-Einstein connections.

## References

The concept was introduced in

• David Mumford, Geometric invariant theory, Ergebnisse Math. Vol 34 Springer (1965)

• F. Takemoto, Stable vector bundles on algebraic surfaces, Nagoya Math. J. 47 (1972) 29-48 (euclid); II, 52 (1973) (euclid)

• David Mumford, John Fogarty, Frances Clare Kirwan, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 34, Springer-Verlag (1965)

Review is in

• Michael Atiyah, Raoul Bott, section 7 of The Yang-Mills equations over Riemann surfaces, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (jstor, lighning summary)

A textbook account is in