Crystals of relative displays and Calabi-Yau threefolds

TL;DR

This paper explores the use of relative displays in crystalline cohomology to establish a deformation theory framework for Calabi-Yau threefolds, connecting p-adic Hodge theory with complex geometry.

Contribution

It introduces a Grothendieck-Messing type theorem for Calabi-Yau threefolds using the crystal of relative displays, advancing deformation theory methods.

Findings

Established a deformation theory framework for Calabi-Yau threefolds.

Connected relative displays with crystalline cohomology in deformation contexts.

Provided new tools for studying Calabi-Yau varieties in p-adic Hodge theory.

Abstract

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 $W\Omega_{X/R}^{\bullet}$ 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.