On complete Kahlerian manifolds endowed with closed conformal vector fields

Let (M)(2n), n > 1, be a complete, noncompact Kahlerian manifold, endowed with a nontrivial closed conformal vector field ? having at least one singular point. Under a reasonable set of conditions, we show that ? has just one singular point p and that (M)\{p} is isometric to a one dimensional cone over a simply connected Sasakian manifold N diffeomorphic to S2n-1.As a straightforward consequence, we conclude that if the addition of a single point to the Kahlerian cone of a (2n - 1)-dimensional Sasakian manifold N has the structure of a complete, noncompact, 2n-dimensional Kahlerian manifold whose metric extends that of the cone, and such that the canonical vector field of the cone extends to it having a singularity at the extra point, then N is isometric to S2n-1, endowed with an appropriate Sasakian structure.