In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory.
Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration.
Stable bundles were defined by David Mumford in Mumford (1963) and later built upon by David Gieseker, Fedor Bogomolov, Thomas Bridgeland and many others.
One of the motivations for analyzing stable vector bundles is their nice behavior in families.
In fact, Moduli spaces of stable vector bundles can be constructed using the Quot scheme in many cases, whereas the stack of vector bundles
is an Artin stack whose underlying set is a single point.
Here's an example of a family of vector bundles which degenerate poorly.
This kind of jumping of numerical invariants does not happen in moduli spaces of stable vector bundles.
[3] A slope of a holomorphic vector bundle W over a nonsingular algebraic curve (or over a Riemann surface) is a rational number μ(W) = deg(W)/rank(W).
If W and V are semistable vector bundles and μ(W) >μ(V), then there are no nonzero maps W → V. Mumford proved that the moduli space of stable bundles of given rank and degree over a nonsingular curve is a quasiprojective algebraic variety.
The cohomology of the moduli space of stable vector bundles over a curve was described by Harder & Narasimhan (1975) using algebraic geometry over finite fields and Atiyah & Bott (1983) using Narasimhan-Seshadri approach.
If X is a smooth projective variety of dimension m and H is a hyperplane section, then a vector bundle (or a torsion-free sheaf) W is called stable (or sometimes Gieseker stable) if for all proper non-zero subbundles (or subsheaves) V of W, where χ 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 ≤.
For bundles on curves the stability defined by slopes and by growth of Hilbert polynomial coincide.
Gieseker stability has an interpretation in terms of geometric invariant theory, while μ-stability has better properties for tensor products, pullbacks, etc.
Let X be a smooth projective variety of dimension n, H its hyperplane section.
A slope of a vector bundle (or, more generally, a torsion-free coherent sheaf) E with respect to H is a rational number defined as where c1 is the first Chern class.
For a vector bundle E the following chain of implications holds: E is μ-stable ⇒ E is stable ⇒ E is semistable ⇒ E is μ-semistable.
Then there exists a unique filtration by subbundles such that the associated graded components Fi := Ei+1/Ei are semistable vector bundles and the slopes decrease, μ(Fi) > μ(Fi+1).
Two vector bundles with isomorphic associated gradeds are called S-equivalent.
On higher-dimensional varieties the filtration also always exist and is unique, but the associated graded components may no longer be bundles.
For bundles of degree 0 projectively flat connections are flat and thus stable bundles of degree 0 correspond to irreducible unitary representations of the fundamental group.
Kobayashi and Hitchin conjectured an analogue of this in higher dimensions.
It 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.
Let X be a projective scheme, d a natural number, E a coherent sheaf on X with dim Supp(E) = d. Write the Hilbert polynomial of E as PE(m) = Σdi=0 αi(E)/(i!)
A coherent sheaf E is semistable if the following two conditions hold:[4] A sheaf is called stable if the strict inequality pF(m) < pE(m) holds for large m. Let Cohd(X) be the full subcategory of coherent sheaves on X with support of dimension ≤ d. The slope of an object F in Cohd may be defined using the coefficients of the Hilbert polynomial as
is called μ-semistable if the following two conditions hold:[5] E is μ-stable if the strict inequality holds for all proper nonzero subobjects of E. Note that Cohd is a Serre subcategory for any d, so the quotient category exists.