On the relative minimal model program for fourfolds in positive and mixed characteristic

We show the validity of two special cases of the four-dimensional minimal model program (MMP) in characteristic 5 ]]>: for contractions to-factorial fourfolds and in families over curves ('semistable MMP'). We also provide their mixed characteristic analogues. As a corollary, we show that liftability of positive characteristic threefolds is stable under the MMP and that liftability of three-dimensional Calabi-Yau varieties is a birational invariant. Our results are partially contingent upon the existence of log resolutions.