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Let $X$ be a connected compact complex manifold. Let $G$ be a complex Lie group acting holomorphically on $X$.
Let $p : E_H \longrightarrow X$ be a holomorphic principal $H$-bundle $X$, where $H$ is a connected complex Lie group. It is
natural to ask when can we lift the $G$-action holomorphically to $E_H$ such that $p : E_H \longrightarrow X$ become a $G$-equivariant
principal $H$-bundle? To answer this question, we define notion of a holomorphic $G$-connection on $E_H$ by suitably modifying
the notion of holomorphic connection on $E_H$, and we use it to give a criterion for lifting the $G$-action on $E_H$ equivariantly.
We also give a necessary and sufficient criterion for existence of holomorphic $G$-connection on $E_H$. |