Akizuki-Nakano vanishing theorem and its application for globally F-split 3-folds

In algebraic geometry, it is important to study spaces (=varieties) which may admit singular points. "Smoothing" is an effective way to deal with singularities. Roughly speaking, a smoothing of a (singular) variety helps us to find another variety which is non-singular and is "similar" to the original variety. Namikawa proved that a complex Fano 3-fold whose singular points are at worst ordinary double points admits a smoothing. In this paper, we give an analogy of this result in the case where the Fano variety is not complex, but is defined over a field of positive characteristic. The key ingredient of the proof is the vanishing of the second cohomology of the tangent space, which is a vector space containing an obstruction to construct a smoothing. We prove a positive characteristic analogue of the Akizuki-Nakano type vanishing for 3-dimensional varieties and as a corollary, we conclude the result about a smoothing.