  In this paper we study the class of compact K\"ahler manifolds with positive
orthogonal Ricci curvature: $Ric^\perp>0$. First we illustrate examples of
K\"ahler manifolds with $Ric^\perp>0$ on K\"ahler C-spaces, and construct ones
on certain projectivized vector bundles. These examples show the abundance of
K\"ahler manifolds which admit metrics of $Ric^\perp>0$. Secondly we prove some
(algebraic) geometric consequences of the condition $Ric^\perp>0$ to illustrate
that the condition is also quite restrictive. Finally this last point is made
evident with a classification result in dimension three and a partial
classification in dimension four.
