Boundedness of weak Fano threefolds with fixed Gorenstein index in positive characteristic

In this paper, we give a partial affirmative answer to the BAB conjecture for 3-folds in characteristic p>5. Specifically, we prove that a set D of weak Fano 3-folds over an uncountable algebraically closed field is bounded, if each element X D satisfies certain conditions regarding the Gorenstein index, a complement and Kodaira type vanishing. In the course of the proof, we also study a uniform lower bound for Seshadri constants of nef and big invertible sheaves on projective 3-folds.