Lenny Taelman: Deformations of Calabi–Yau varieties in mixed characteristic

Abstract: We study deformations of smooth projective varieties with trivial canonical bundle in positive and mixed characteristic. We show that (under suitable hypotheses) these are unobstructed. This is an analogue to the Bogomolov–Tian–Todorov theorem (in characteristic zero). We also show that “ordinary” varieties with trivial canonical bundle admit a preferred “canonical lift” to characteristic zero. This generalizes results of Serre–Tate (for abelian varieties) and Nygaard and Deligne (for K3 surfaces). Our proofs rely in an essential way on derived deformation theory as pioneered by Pridham and Lurie.

This is joint work with Lukas Brantner.