Projectively induced K\"ahler cones over regular Sasakian manifolds

Motivated by a conjecture in [9] we prove that the K\"ahler cone over a regular complete Sasakian manifold is Ricci-flat and projectively induced if and only if it is flat. We also obtain that, up to $\mathcal D_a$-homothetic transformations, K\"ahler cones over homogeneous compact Sasakian manifolds are projectively induced. As main tool we provide a relation between the K\"ahler potentials of the transverse K\"ahler metric and of the cone metric.