The conformal submersions of Kählerian manifolds. I

The Kählerian manifolds permitting holomorphic Riemannian submersions are necessary reducible. The paper therefore analyzes mainly the conformal submersions which are not Riemannian. A description is obtained for the structure of the curvature tensor of the Kählerian manifold $E$ permitting a holomorphic conformal submersion onto another Kählerian manifold. The fibers of a submersion are assumed to be totally geodesic. The structure of the Kählerian metric of the manifold $E$ is described for this type of submersions whose fibers have the complex dimension equal to 1. Particular examples will be given. A method will be proposed later on to construct bundles whose projection is a holomorphic conformal (non-Riemannian) submersion with vertical exponent of conformality and totally geodesic fibers. The method allows construction of the complete (including compact) Kählerian fiber space with the project of the above indicated type. It will be shown that the Hodge base is the necessary and sufficient condition for existence of such bundles.