Stable vector bundle
Stable vector bundles over curves
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.
Stable vector bundles over projective varieties
If X is a smooth projective variety of dimension n and H is a hyperplane section, then a vector bundle (or torsionfree sheaf) W is called stable if
for all proper non-zero subbundles (or subsheaves) V of W, where denotes the Euler characteristic of an algebraic vector bundle and the vector bundle means the n-th twist of V by H. W is called semistable if the above holds with < replaced by ≤.
There are also other variants in the literature: cf. this thesis p.29.
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- Donaldson, S. K. (1985), "Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles", Proceedings of the London Mathematical Society. Third Series 50 (1): 1–26, doi:10.1112/plms/s3-50.1.1, ISSN 0024-6115, MR 765366
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- Narasimhan, M. S.; Seshadri, C. S. (1965), "Stable and unitary vector bundles on a compact Riemann surface", Annals of Mathematics. Second Series (The Annals of Mathematics, Vol. 82, No. 3) 82 (3): 540–567, doi:10.2307/1970710, ISSN 0003-486X, JSTOR 1970710, MR 0184252