Diameter theorems on K\"ahler and quaternionic K\"ahler manifolds under a positive lower curvature bound

We define the orthogonal Bakry-\'Emery tensor as a generalization of the orthogonal Ricci curvature, and then study diameter theorems on K\"ahler and quaternionic K\"ahler manifolds under positivity assumption on the orthogonal Bakry-\'Emery tensor. Moreover, under such assumptions on the orthogonal Bakry-\'Emery tensor and the holomorphic or quaternionic sectional curvature on a K\"ahler manifold or a quaternionic K\"ahler manifold respectively, the Bonnet-Myers type diameter bounds are sharper than in the Riemannian case.