Invariant connections and invariant holomorphic bundles on homogeneous manifolds

TL;DR

This paper provides explicit classification theorems for invariant connections, gauge classes, and holomorphic bundles on homogeneous manifolds with a transitive Lie group action, under certain technical assumptions.

Contribution

It offers new explicit classification results for invariant geometric structures on homogeneous manifolds, connecting them to finite dimensional quotient spaces.

Findings

Classified invariant $K$-connections on $X$

Classified gauge classes of $K$-connections

Classified invariant holomorphic bundles with reductions

Abstract

Let $X$ be a differentiable manifold endowed with a transitive action $\alpha:A\times X\longrightarrow X$ 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 $Q\longrightarrow X$ and a $K$-reduction $P$ of $Q$ (when $X$ has an $\alpha$-invariant complex structure). {enumerate}