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

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