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. 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 , who showed that in this case a vector bundle is stable if and only if it has an irreducible Hermitian-Einstein connection.
for all proper non-zero subbundles (or subsheaves) V of W, and is semistable if the above holds with < replaced by ≤.