Stable vector bundle

In mathematics, a stable vector bundle is a vector bundle that is stable in the sense of geometric invariant theory.

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. 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 connections. 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; 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