nLab
stable vector bundle

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

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

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

Properties

Narasimhan-Seshadri theorem

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

• Daniel Huybrechts, Lehn, The Geometry of the Moduli Space of Sheaves

See also

• Paolo de Bartolomeis, Gang Tian, Stability of complex vector bundles, Journal of Differential Geometry, Vol. 43, No. 2 (1996) (pdf)

• Wei-Ping Li, Zhenbo Qin, Stable vector bundles on algebraic surfaces (pdf)