Stable principal bundle

In mathematics, and especially differential geometry and algebraic geometry, a stable principal bundle is a generalisation of the notion of a stable vector bundle to the setting of principal bundles. The concept of stability for principal bundles was introduced by Annamalai Ramanathan for the purpose of defining the moduli space of G-principal bundles over a Riemann surface, a generalisation of earlier work by David Mumford and others on the moduli spaces of vector bundles.
Many statements about the stability of vector bundles can be translated into the language of stable principal bundles. For example, the analogue of the Kobayashi–Hitchin correspondence for principal bundles, that a holomorphic principal bundle over a compact Kähler manifold admits a Hermite–Einstein connection if and only if it is polystable, was shown to be true by Boudjemâa Anchouche and Indranil Biswas.
Definition.
The essential definition of stability for principal bundles was made by Ramanathan, but applies only to the case of Riemann surfaces. In this section we state the definition as appearing in the work of Anchouche and Biswas which is valid over any Kähler manifold, and indeed makes sense more generally for algebraic varieties. This reduces to Ramanathan's definition in the case the manifold is a Riemann surface.
Let formula_1 be a connected reductive algebraic group over the complex numbers formula_2. Let formula_3 be a compact Kähler manifold of complex dimension formula_4. Suppose formula_5 is a holomorphic principal formula_1-bundle over formula_7. Holomorphic here means that the transition functions for formula_8 vary holomorphically, which makes sense as the structure group is a complex Lie group. The principal bundle formula_8 is called stable (resp. semi-stable) if for every reduction of structure group formula_10 for formula_11 a maximal parabolic subgroup where formula_12 is some open subset with the codimension formula_13, we have 
Here formula_15 is the relative tangent bundle of the fibre bundle formula_16 otherwise known as the vertical bundle of formula_17. Recall that the degree of a vector bundle (or coherent sheaf) formula_18 is defined to be 
where formula_20 is the first Chern class of formula_21. In the above setting the degree is computed for a bundle defined over formula_22 inside formula_7, but since the codimension of the complement of formula_22 is bigger than two, the value of the integral will agree with that over all of formula_7.
Notice that in the case where formula_26, that is where formula_7 is a Riemann surface, by assumption on the codimension of formula_22 we must have that formula_29, so it is enough to consider reductions of structure group over the entirety of formula_7, formula_31.
Relation to stability of vector bundles.
Given a principal formula_1-bundle for a complex Lie group formula_1 there are several natural vector bundles one may associate to it. 
Firstly if formula_34, the general linear group, then the standard representation of formula_35 on formula_36 allows one to construct the associated bundle formula_37. This is a holomorphic vector bundle over formula_7, and the above definition of stability of the principal bundle is equivalent to slope stability of formula_39. The essential point is that a maximal parabolic subgroup formula_40 corresponds to a choice of flag formula_41, where formula_42 is invariant under the subgroup formula_43. Since the structure group of formula_8 has been reduced to formula_43, and formula_43 preserves the vector subspace formula_47, one may take the associated bundle formula_48, which is a sub-bundle of formula_39 over the subset formula_12 on which the reduction of structure group is defined, and therefore a subsheaf of formula_39 over all of formula_7. It can then be computed that
where formula_54 denotes the slope of the vector bundles.
When the structure group is not formula_55 there is still a natural associated vector bundle to formula_8, the adjoint bundle formula_57, with fibre given by the Lie algebra formula_58 of formula_1. The principal bundle formula_8 is semistable if and only if the adjoint bundle formula_57 is slope semistable, and furthermore if formula_8 is stable, then formula_57 is slope polystable. Again the key point here is that for a parabolic subgroup formula_11, one obtains a parabolic subalgebra formula_65 and can take the associated subbundle. In this case more care must be taken because the adjoint representation of formula_1 on formula_58 is not always faithful or irreducible, the latter condition hinting at why stability of the principal bundle only leads to "polystability" of the adjoint bundle (because a representation that splits as a direct sum would lead to the associated bundle splitting as a direct sum).
Generalisations.
Just as one can generalise a vector bundle to the notion of a Higgs bundle, it is possible to formulate a definition of a principal formula_1-Higgs bundle. The above definition of stability for principal bundles generalises to these objects by requiring the reductions of structure group are compatible with the Higgs field of the principal Higgs bundle. It was shown by Anchouche and Biswas that the analogue of the nonabelian Hodge correspondence for Higgs vector bundles is true for principal formula_1-Higgs bundles in the case where the base manifold formula_3 is a complex projective variety.