Crystals of relative displays and Grothendieck-Messing Deformation 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 of Langer-Zink, and can be thought of as a kind of mixed characteristic Hodge structure.

In this thesis, we show that under certain geometric conditions, deforming X over PD-thickenings of R gives a crystal of relative displays. We then apply the crystal of relative displays to prove Grothendieck-Messing type results for the deformation theory of Calabi-Yau threefolds. We also show that primitive crystalline cohomology often carries a display-structure, and we prove a Grothendieck-Messing type result for the deformation theory of smooth cubic fourfolds in terms of the crystal of relative displays on primitive crystalline cohomology. Finally, we investigate the deformation theory of ordinary smooth cubic fourfolds in terms of the displays on the cohomology of their Fano schemes of lines.