  Let $X$ be a complete variety over an algebraically closed field $k$ of
characteristic zero, equipped with an action of an algebraic group $G$. Let $H$
be a reductive group. We study the notion of $G$-connection on a principal
$H$-bundle. We give necessary and sufficient criteria for the existence of
$G$-connections extending the Atiyah-Weil type criterion for holomorphic
connections obtained by Azad and Biswas. We also establish a relationship
between the existence of $G$-connection and equivariant structure on a
principal $H$-bundle, under the assumption that $G$ is semisimple and simply
connected. These results have been obtained by Biswas et al. when the
underlying variety is smooth.
