Conification construction for Kähler manifolds and its application in c-projective geometry

Two Kähler metrics on a complex manifold are called c-projectively equivalent if their J-planar curves coincide. Such curves are defined by the property that the acceleration is complex proportional to the velocity. The degree of mobility of a Kähler metric is the dimension of the space of metrics that are c-projectively equivalent to it. We give the list of all possible values of the degree of mobility of a simply connected Kähler manifold by reducing the problem to the study of parallel Hermitian (0,2)-tensors on the conification of the manifold. We also describe all such values for a Kähler-Einstein metric. We apply these results to describe all possible dimensions of the space of essential c-projective vector fields of Kähler and Kähler-Einstein metrics. We also show that two c-projectively equivalent Kähler-Einstein metrics (of arbitrary signature) on a closed manifold have constant holomorphic curvature or are affinely equivalent.