Canonical Bochner-Kähler potentials via the Sasaki-Kähler correspondence

Abstract: We introduce a new approach to the study of Bochner-Kähler manifolds, based on normal form techniques in CR geometry. As observed by Webster, Bochner-Kähler manifolds are intimately related to spherical Sasakian manifolds. Our approach utilises this relationship as well as normal forms for spherical Sasakian manifolds (known as rigid spheres). In this setting potentials of Bochner-Kähler manifolds are nothing but the defining equations of rigid spheres in rigid normal form. This leads to a canonical formula for Bochner-Kähler potentials, as well as streamlined proofs of known results on the structure and the symmetries of Bochner-Kähler manifolds. Our approach also provides a host of examples in terms of an implicit formula of the corresponding Kähler potentials.