Principal bundles with holomorphic connections over a K?hler Calabi-Yau manifold

We prove that any holomorphic vector bundle admitting a holomorphic connection, over a compact Kahler Calabi-Yau manifold, also admits a flat holomorphic connection. This addresses a particular case of a question asked by Atiyah and generalizes a result previously obtained in [6] for simply connected compact Kahler Calabi-Yau manifolds. We give some applications of it in the framework of Cartan geometries and foliated Cartan geometries on Kahler Calabi-Yau manifolds.(c) 2023 Elsevier B.V. All rights reserved.