A classical theorem of Thierry Aubin states that:

Theorem (Aubin, T. 1979): If the Ricci curvature of a compact Riemannian manifold is non-negative and positive at a point, then the manifold carries a metric of positive Ricci curvature.

In the study of structures on manifolds (such as Hermitian, Kaehlerian, symplectic,...) is the above theorem true? i.e.

Question: Does "the Ricci curvature of a compact Hermitian manifold is non-negative and positive at a point", imply "the manifold carries a Hermitian metric of positive Ricci curvature"?

Your suggestions will be appreciated.