This paper by M. F. Atiyah explores the theory of complex analytic connections in fiber bundles, a topic that differs significantly from the differentiable case. In the differentiable setting, connections always exist but may not be integrable, while in the complex analytic setting, connections may not exist at all. The author introduces obstructions to the existence of connections and integrability, showing that these obstructions generate the characteristic cohomology ring of the bundle in many important cases. This approach provides a purely cohomological definition of the characteristic ring, which is canonical and independent of the choice of connection. The paper also discusses the application of these results to the problem of characterizing fiber bundles arising from representations of the fundamental group, as studied by Weil. The author defines and analyzes various concepts such as vector bundles, coherent sheaves, and extensions, and develops the theory of complex analytic connections, including the construction of the curvature and the relationship between connections and Chern classes. The paper concludes with a detailed treatment of Chern classes for split vector bundles and general vector bundles over compact Kähler manifolds.This paper by M. F. Atiyah explores the theory of complex analytic connections in fiber bundles, a topic that differs significantly from the differentiable case. In the differentiable setting, connections always exist but may not be integrable, while in the complex analytic setting, connections may not exist at all. The author introduces obstructions to the existence of connections and integrability, showing that these obstructions generate the characteristic cohomology ring of the bundle in many important cases. This approach provides a purely cohomological definition of the characteristic ring, which is canonical and independent of the choice of connection. The paper also discusses the application of these results to the problem of characterizing fiber bundles arising from representations of the fundamental group, as studied by Weil. The author defines and analyzes various concepts such as vector bundles, coherent sheaves, and extensions, and develops the theory of complex analytic connections, including the construction of the curvature and the relationship between connections and Chern classes. The paper concludes with a detailed treatment of Chern classes for split vector bundles and general vector bundles over compact Kähler manifolds.