This paper by M. F. Atiyah explores complex analytic connections in complex analytic fibre bundles, contrasting them with the differentiable case. In the differentiable setting, connections always exist but may not be integrable, while in the complex analytic case, connections may not exist at all. The paper shows that, in a large class of important cases, the obstruction to the existence of a complex analytic connection generates the characteristic cohomology ring of the bundle. This leads to a purely cohomological definition of the characteristic ring, which has advantages over the differentiable approach, including being canonical and not depending on arbitrary choices of connections. The paper develops the ideas of vector bundles, sheaves, and complex analytic connections in detail, applying them to a problem first studied by Weil. It introduces the concept of characteristic rings and their relation to Chern classes, and discusses the behavior of these structures under various operations such as tensor products and direct sums. The paper also examines the relationship between the characteristic ring of a principal bundle and the characteristic ring of the associated vector bundle. It shows that the obstruction to the existence of a complex analytic connection in a principal bundle is related to the obstruction in the associated vector bundle, and that these obstructions generate the characteristic cohomology ring of the bundle. The paper concludes with a discussion of the implications of these results for the existence of complex analytic connections in terms of the characteristic ring, and notes that the converse of a key theorem is not generally true. The paper provides a comprehensive treatment of complex analytic fibre bundles, their connections, and the associated cohomology theories.This paper by M. F. Atiyah explores complex analytic connections in complex analytic fibre bundles, contrasting them with the differentiable case. In the differentiable setting, connections always exist but may not be integrable, while in the complex analytic case, connections may not exist at all. The paper shows that, in a large class of important cases, the obstruction to the existence of a complex analytic connection generates the characteristic cohomology ring of the bundle. This leads to a purely cohomological definition of the characteristic ring, which has advantages over the differentiable approach, including being canonical and not depending on arbitrary choices of connections. The paper develops the ideas of vector bundles, sheaves, and complex analytic connections in detail, applying them to a problem first studied by Weil. It introduces the concept of characteristic rings and their relation to Chern classes, and discusses the behavior of these structures under various operations such as tensor products and direct sums. The paper also examines the relationship between the characteristic ring of a principal bundle and the characteristic ring of the associated vector bundle. It shows that the obstruction to the existence of a complex analytic connection in a principal bundle is related to the obstruction in the associated vector bundle, and that these obstructions generate the characteristic cohomology ring of the bundle. The paper concludes with a discussion of the implications of these results for the existence of complex analytic connections in terms of the characteristic ring, and notes that the converse of a key theorem is not generally true. The paper provides a comprehensive treatment of complex analytic fibre bundles, their connections, and the associated cohomology theories.