On a Kahler manifold there is a clear connection between the  complex geometry and underlying Riemannian geometry, which can be used to characterize the Kahler condition. While such a link is not as clear in the non-Kahler setting, one can seek to understand these  characterizations as specific instances of a more general type.  I will address such questions from the perspective  of relationships between the Chern and Riemannian scalar curvatures. This is joint work with Michael Dabkowski.