On Homogeneous K\"ahler Manifolds
The cone $M \times \mathbb{R}_+$ over a Sasakian manifold $M$ is equipped with a canonical K\"ahler structure with specific homogeneity properties with respect to the $\mathbb{R}_+$ coordinate. This K\"ahler structure completely encodes the underlying Sasakian structure. Recently, Grabowski, Grabowska and Mohseni provided a broader conceptual framework for this phenomenon via homogeneous K\"ahler structures, i.e. K\"ahler structures on a principal $\mathbb{R}^\times$-bundle $P$ satisfying similar homogeneity properties. This approach successfully extends Sasakian geometry from cooriented contact manifolds (where $P$ is a trivial principal bundle) to non-necessarily coorientable contact structures (where $P$ is non-necessarily trivial). Homogeneous K\"ahler structures are genuinely more general than Sasakian structures and this note precisely characterizes the extent of this generalization. This is achieved through a detailed analysis of all the involved compatibilities in terms of the line bundle tautologically associated to $P$. We also show that modifying the homogeneity condition on the K\"ahler structure allows this framework to encompass co-K\"ahler structures and a natural generalization of those as well.