  In this paper, we prove that a weak form of the Akizuki-Nakano vanishing
theorem holds on globally $F$-split 3-folds. Making use of this vanishing
theorem, we study deformations of globally $F$-split Fano 3-folds and the
Kodaira vanishing theorem for thickenings of locally complete intersection
globally $F$-regular 3-folds.
