Stable principal bundles and reduction of structure group

Abstract

Let E_G be a stable principal G.bundle over a compact connected Kahler manifold, where G is a connected reductive linear algebraic group defined over C. Let H ⊂ G be a complex reductive subgroup which is not necessarily connected, and let E_H ⊂ E_G be a holomorphic reduction of structure group. We prove that E_H is preserved by the Einstein.Hermitian connection on E_G. Using this we show that if E_H is a minimal reductive reduction in the sense that there is no complex reductive proper subgroup of H to which E_H admits a holomorphic reduction of structure group, then E_H is unique in the following sense: For any other minimal reductive reduction (H'E ,EH' ) of EG, there is some element g ∈ G such that H'= g^-1H_g and E_H' = E_Hg. As an application, we give an affirmative answer to a question of Balaji and Kollar.