nLab
stable vector bundle

Contents

Definition

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

deg(K)rank(K)<deg(E)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

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)

Revised on November 7, 2012 19:49:04 by Urs Schreiber (82.169.65.155)