Characterising canonical lifts of ordinary Calabi-Yau varieties

We study Brantner-Taelman's canonical lifts of ordinary Calabi-Yau varieties and characterise their (quasi)canonical lifts in terms of rational $p$-adic Hodge theoretic data. We also show that these canonical lifts need not be of CM type and that the associated local Galois representations need not split as sums of twists of unramified representations, in contrast to classical results for abelian varieties and K3 surfaces.