A theorem on compact locally conformal kahler manifolds

We prove that a compact locally conformai Kahler manifold which satisfies either: (1) it has nonpositive conformai invariant μ [2] and its local conformai Kahler metrics have nonnegative scalar curvature or (2) its local conformai Kähler (l.c.K.) metrics have a positive or negative definite Ricci form is a Kähler manifold. We conjecture that every compact l.c.K. manifold which satisfies all the topological restrictions of a Kahler manifold admits some Kahler metric.