Manifolds with positive orthogonal Ricci curvature

In this paper we study the class of compact K\"ahler manifolds with ${\rm Ric}^\perp>0$. First we illustrate examples of K\"ahler manifolds with ${\rm Ric}^\perp>0$ 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 ${\rm Ric}^\perp>0$. Secondly we prove some (algebraic) geometric consequences of the condition ${\rm Ric}^\perp>0$ 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.