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 in the case of projective manifolds by Subramanian and Ramanathan, and for arbitrary compact Kähler manifolds by Anchouche and 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 G {\displaystyle G} be a connected reductive algebraic group over the complex numbers C {\displaystyle \mathbb {C} } . Let ( X , ω ) {\displaystyle (X,\omega )} be a compact Kähler manifold of complex dimension n {\displaystyle n} . Suppose P → X {\displaystyle P\to X} is a holomorphic principal G {\displaystyle G} -bundle over X {\displaystyle X} . Holomorphic here means that the transition functions for P {\displaystyle P} vary holomorphically, which makes sense as the structure group is a complex Lie group. The principal bundle P {\displaystyle P} is called stable (resp. semi-stable) if for every reduction of structure group σ : U → P / Q {\displaystyle \sigma :U\to P/Q} for Q ⊂ G {\displaystyle Q\subset G} a maximal parabolic subgroup where U ⊂ X {\displaystyle U\subset X} is some open subset with the codimension codim ( X ∖ U ) ≥ 2 {\displaystyle \operatorname {codim} (X\backslash U)\geq 2} , we have
deg σ ∗ T rel P / Q > 0 ( resp. ≥ 0 ) . {\displaystyle \deg \sigma ^{*}T_{\operatorname {rel} }P/Q>0\quad ({\text{resp. }}\geq 0).}
Here T rel P / Q {\displaystyle T_{\operatorname {rel} }P/Q} is the relative tangent bundle of the fibre bundle P / Q | U → U {\displaystyle \left.P/Q\right|_{U}\to U} otherwise known as the vertical bundle of T ( P / Q | U ) {\displaystyle T(\left.P/Q\right|_{U})} . Recall that the degree of a vector bundle (or coherent sheaf) F → X {\displaystyle F\to X} is defined to be
deg ( F ) := ∫ X c 1 ( F ) ∧ ω n − 1 , {\displaystyle \operatorname {deg} (F):=\int _{X}c_{1}(F)\wedge \omega ^{n-1},}
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