On the Algebraic Holonomy of Stable Principal Bundles

Let EG be a stable principal G–bundle over a compact connected Kähler 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 EH ⊂ EG be a holomorphic reduction of structure group. We prove that EH is preserved by the Einstein–Hermitian connection on EG. Using this we show that if EH is a minimal reductive reduction, which means that there is no complex reductive proper subgroup of H to which EH admits a holomorphic reduction of structure group, then EH is unique in the following sense: For any other minimal reduction of structure group (H ′ , EH′ ) of EG to some reductive subgroup H , there is some element g ∈ G such that H ′ = gHg and EH′ = EHg. As an application, we show the following: Let M be a simply connected, irreducible smooth complex projective variety of dimension n such that the Picard number of M is one. If the canonical line bundle KM is ample, then the algebraic holonomy of the holomorphic tangent bundle T M is GL(n, C). If K M is ample, the rank of the Picard group of M is one, the biholomorphic automorphism group of M is finite, and M admits a Kähler–Einstein metric, then the algebraic holonomy of T M is GL(n, C). These answer some questions posed in [BK].