Principal bundles on compact complex manifolds with trivial tangent bundle

Abstract

Let G be a connected complex Lie group and $${\Gamma \subset G}$$ a cocompact lattice. Let H be a complex Lie group. We prove that a holomorphic principal H-bundle E H over G/Γ admits a holomorphic connection if and only if E H is invariant. If G is simply connected, we show that a holomorphic principal H-bundle E H over G/Γ admits a flat holomorphic connection if and only if E H is homogeneous.