G-connections on principal bundles over complete G-varieties

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 AtiyahWeil 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. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.