Stable Vector Bundles Over Curves
A bundle W over an algebraic curve (or over a Riemann surface) is stable if and only if
for all proper non-zero subbundles V of W and is semistable if
for all proper non-zero subbundles V of W. Informally this says that a bundle is stable if it is "more ample" than any proper subbundle, and is unstable if it contains a "more ample" subbundle. The moduli space of stable bundles of given rank and degree is an algebraic variety.
Narasimhan & Seshadri (1965) showed that stable bundles on projective nonsingular curves are the same as those that have projectively flat unitary irreducible connections; these correspond to irreducible unitary representations of the fundamental group. Kobayashi and Hitchin conjectured an analogue of this in higher dimensions; this was proved for projective nonsingular surfaces by Donaldson (1985), who showed that in this case a vector bundle is stable if and only if it has an irreducible Hermitian–Einstein connection.
The cohomology of the moduli space of stable vector bundles over a curve was described by Harder & Narasimhan (1975) and Atiyah & Bott (1983).
Read more about this topic: Stable Vector Bundle
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