In the context of mirror symmetry, the moduli space of complex Calabi-Yau varieties acquires canonical local coordinates near a "large complex structure limit point". In characteristic p geometry, the formal deformation space of an ordinary Calabi-Yau variety tends to have such canonical coordinates ("Serre-Tate parameters") as well. As observed e.g. by Jan Stienstra, these situations are formally very similar, and one would like to compare the two when both make sense. A framework for doing this could be supplied by a version of Serre-Tate theory for log Calabi-Yau varieties. In my talk, I will describe the first step in this direction; a construction of canonical liftings modulo $p^2$ of certain log schemes. I will link this to a question of Keel describing global moduli of maximal log Calabi-Yau pairs.