Crystals of relative displays and Calabi-Yau threefolds

Displays can be thought of as relative versions of Fontaine’s notion of strongly divisible lattice from integral $p$-adic Hodge theory. In favourable circumstances, the crystalline cohomology of a smooth projective $R$-scheme $X$ is endowed with a display-structure coming from complexes associated with the relative de Rham-Witt complex $W{\mathrm{\Omega }}_{X\mathrm{/}R}^{\bullet }$ of [LZ04]. In this article, we use the crystal of relative displays of [GL21] to prove a Grothendieck-Messing type result for the deformation theory of Calabi-Yau threefolds.