A classical theorem of Thierry Aubin states that:

>**Theorem ([Aubin, T. 1979][1]):** 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.


  [1]: https://books.google.com/books?id=iM7uCAAAQBAJ&lpg=PP1&pg=PA344#v=onepage&q&f=false