Stable vector bundle

In mathematics, a stable vector bundle is a vector bundle that is stable in the sense of geometric invariant theory. They were defined by Mumford (1963).

Stable vector bundles over curves

A bundle W over an algebraic curve (or over a Riemann surface) is stable if and only if

$\displaystyle {\frac {\deg(V)}{{\hbox{rank}}(V)}}<{\frac {\deg(W)}{{\hbox{rank}}(W)}}$

for all proper non-zero subbundles V of W and is semistable if

$\displaystyle {\frac {\deg(V)}{{\hbox{rank}}(V)}}\leq {\frac {\deg(W)}{{\hbox{rank}}(W)}}$

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).

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

${\frac {\chi (V(nH))}{{\hbox{rank}}(V)}}<{\frac {\chi (W(nH))}{{\hbox{rank}}(W)}}{\text{ for }}n{\text{ large}}$

for all proper non-zero subbundles (or subsheaves) V of W, and is semistable if the above holds with < replaced by ≤.

References

• 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, ISSN 0080-4614, MR702806
• 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, MR765366
• Friedman, Robert (1998), Algebraic surfaces and holomorphic vector bundles, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98361-5, MR1600388
• Harder, G.; Narasimhan, M. S. (1975), "On the cohomology groups of moduli spaces of vector bundles on curves", Mathematische Annalen, 212: 215–248, doi:10.1007/BF01357141, ISSN 0025-5831, MR0364254
• Mumford, David (1963), "Projective invariants of projective structures and applications", Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 526–530, MR0175899
• Mumford, David; Fogarty, J.; Kirwan, F. (1994), Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-56963-3, MR1304906 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, 82: 540–567, ISSN 0003-486X, MR0184252