# 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

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.

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

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.

## See also

## 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, 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