Comparison and vanishing theorems for K\"ahler manifolds

In this paper, we consider orthogonal Ricci curvature $Ri{c}^{\perp }$ for K"ahler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate their subtle relationship.