Localization of Atiyah classes

We construct the Atiyah classes of holomorphic vector bundles using (1,0)-connections and developing a Chern–Weil type theory, allowing us to effectively compare Chern and Atiyah forms. Combining this point of view with the Čech–Dolbeault cohomology, we prove several results about vanishing and localization of Atiyah classes, and give some applications. In particular, we prove a Bott type vanishing theorem for (not necessarily involutive) holomorphic distributions. As an example we also present an explicit computation of the residue of a singular distribution on the normal bundle of an invariant submanifold that arises from the Camacho–Sad type localization.