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.
- Atiyah, Michael Francis; Bott, Raoul (1983), "The Yang-Mills equations over Riemann surfaces", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 308 (1505): 523–615, doi:10.1098/rsta.1983.0017, ISSN 0080-4614, JSTOR 37156, MR 702806
- 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
- Friedman, Robert (1998), Algebraic surfaces and holomorphic vector bundles, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98361-5, MR 1600388
- Harder, G.; Narasimhan, M. S. (1975), "On the cohomology groups of moduli spaces of vector bundles on curves", Mathematische Annalen, 212 (3): 215–248, doi:10.1007/BF01357141, ISSN 0025-5831, MR 0364254
- Mumford, David (1963), "Projective invariants of projective structures and applications", Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 526–530, MR 0175899
- Mumford, David; Fogarty, J.; Kirwan, F. (1994), Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-56963-3, MR 1304906 especially appendix 5C.
- 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