In this talk, we discuss the Akizuki-Nakano type vanishing theorem and its application in positive characteristic. By Raynaud's result, Akizuki-Nakano vanishing theorem holds on a smooth globally F-split variety if the characteristic is larger than or equal to the dimension. However, very little is known for singular varieties. We first show a weak form of the Akizuki-Nakano vanishing theorem on (possibly singular) globally F-split 3-folds. As applications, we prove a Lefschetz hyperplane type result and results on deformation theory on globally F-split Fano 3-folds with isolated complete intersection singularities. In particular, I explain that a globally F-split Fano 3-fold with ordinary double point admits a smoothing and is liftable to characteristic 0 under a mild condition, which can be seen as a positive characteristic analogue of Namikawa's result.