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

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

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, where $\chi$ denotes the Euler characteristic of an algebraic vector bundle and the vector bundle $V(nH)$ 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

• Kobayashi–Hitchin correspondence
• Corlette–Simpson correspondence

References

• especially appendix 5C.