Examples of a non-vanishing conjecture of Kawamata

Let M be a compact complex manifold with a positive holomorphic line bundle L, and K be its canonical line bundle. We give some sufficient conditions for the non-vanishing of H⁰ (M, K + L). We also show that the criterion can be applied to interesting classes of examples including all compact locally hermitian symmetric spaces of non-compact type, Mostow-Siu [10] surfaces, Kähler threefolds given by Deraux [3] and examples of Zheng [17].