geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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.
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
between their slopes holds, i.e. if the inequality
holds between the fractions of degree and rank of the vector bundles holds.
e.g. (Huybrechts-Lehn 96, bottom of p. 24)
e.g. (Huybrechts-Lehn 96, def. 1.2.4, def. 1.2.12)
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)
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.
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.
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
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)