Invariant connections and invariant holomorphic bundles on homogeneous manifolds

Abstract

Let $X$ be a differentiable manifold endowed with a transitive action $\alpha :A\times X⟶X$ of a Lie group $A$. Let $K$ be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects: {enumerate} equivalence classes of $\alpha$-invariant $K$-connections on $X$, $\alpha$-invariant gauge classes of $K$-connections on $X$, and $\alpha$-invariant isomorphism classes of pairs $(Q,P)$ consisting of a holomorphic $K^\C$-bundle $Q⟶X$ and a $K$-reduction $P$ of $Q$ (when $X$ has an $\alpha$-invariant complex structure). {enumerate}

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