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 harvtxt|Mumford|1963

table vector bundles over curves

A bundle "W" over an algebraic curve (or over a Riemann surface) is stable if and only if

: $displaystylefrac{deg(V)}{hbox{rank}(V)} < frac{deg(W)}{hbox{rank}(W)}$

for all proper non-zero subbundles "V" of "W" and is semistable if

: $displaystylefrac{deg(V)}{hbox{rank}(V)} le frac{deg(W)}{hbox{rank}(W)}$

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.

harvtxt|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 harvtxt|Donaldson|1985, who showed that in this case a vector bundle is stable if and only if it has an irreducible Hermitian-Einstein connection.

table 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

: $frac{chi(V(nH))}{hbox{rank}(V)} < frac{chi(W(nH))}{hbox{rank}(W)} ext{ for }n ext{ large}$

for all proper non-zero subbundles (or subsheaves) "V" of "W", and is semistable if the above holds with < replaced by ≤.

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