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 Cech-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.