Stable vector bundle

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

A bundle W over an algebraic curve (or 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.

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
• Harder, G.; Narasimhan, M. S. (75), "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 {{citation}}: Check date values in: |year= (help); Unknown parameter |month= ignored (help)CS1 maint: year (link)
• 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