  We study deformations of Calabi-Yau varieties in characteristic $p$ using
techniques from derived algebraic geometry. We prove a mixed characteristic
analogue of the Bogomolov-Tian-Todorov theorem (which states that Calabi-Yau
varieties in characteristic $0$ are unobstructed), and we show that ordinary
Calabi-Yau varieties admit canonical lifts to characteristic $0$, generalising
the Serre-Tate theorem on ordinary abelian varieties.
