The Atiyah class of generalized holomorphic vector bundles

Abstract

For a generalized holomorphic vector bundle, we introduce the Atiyah class, which is the obstruction of the existence of generalized holomorphic connections on this bundle. Similar to the holomorphic case, such Atiyah classes can be defined by three approaches: the Čech cohomology, the extension class of the first jet bundle as well as the Lie pair.