Deformations of Calabi-Yau varieties in mixedcharacteristic

A smooth projective variety X is said to be Calabi-Yau if itscanonical bundle is trivial. I will discuss joint work with Lukas Brantner, inwhich we use derived algebraic geometry to study deformations of Calabi-Yauvarieties in characteristic p.

We prove a positive characteristic analogue of the Bogomolov-Tian-Todorovtheorem (which states that deformations of Calabi-Yau varieties incharacteristic 0 are unobstructed), and show that 'ordinary' Calabi-Yauvarieties admit canonical lifts to characteristic zero (generalising earlierresults of Serre-Tate for abelian varieties, and Deligne and Nygaard for K3surfaces).