A vector bundle over a Riemann surface is stable (resp. semistable) if the ratio of its degree and its rank is bigger (resp. bigger or equal) than this ratio for any of its subbundles.
Many references are at moduli space of bundles. The notion is introduced by David Mumford in the context of GIT in 1960-s:
- D. Mumford, J. Fogarty, F. Kirwan, (1994), Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 34, 3rd ed. Springer 1994
- G. Faltings, Stable G-bundles and projective connections, J. Alg. Geometry , 2 (1993) pp. 507–568 MR1211997 Zbl 0790.14019