nLab
stable vector bundle

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

μ(K)<μ(E) \mu(K) \lt \mu(E)

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

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

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

e.g. (Huybrechts-Lehn 96, bottom of p. 24)

Over a general (Noetherian) scheme

e.g. (Huybrechts-Lehn 96, def. 1.2.4, def. 1.2.12)

Examples

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

e.g. (Huybrechts-Lehn 96, example 1.2.10)

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

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)

  • Alfonso Zamora Saiz, On the stability of vector bundles, Master thesis 2009 (pdf)

Revised on July 14, 2014 08:01:07 by Urs Schreiber (89.204.135.179)