Serre-Tate theory for Calabi-Yau varieties

Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the moduli space with a Frobenius lifting and canonical multiplicative coordinates. I will discuss a project, joint with Maciej Zdanowicz (EPFL), whose aim is to construct canonical liftings modulo $p^2$ of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two.