bundles

complex geometry

# Contents

## Definition

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

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

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

## Properties

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

## References

The notion was introduced in

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

and

• F. Takemoto, Stable vector bundles on algebraic surfaces, Nagoya Math. J. 47 (1973) and 52 (1973) (Euclid)

A textbook account is in