Speaker : Lukas Brantner (University of Oxford) 

Title : On deformations and lifts of Calabi-Yau varieties in characteristic $p$

Abstract:  A smooth projective variety $Z$ is said to be Calabi-Yau if its canonical bundle is trivial. I will discuss recent joint work with Taelman, in which we use derived algebraic geometry to study how Calabi-Yau varieties in characteristic p deform. More precisely, we show that if $Z$ has degenerating Hodge-de Rham spectral sequence and torsion-free crystalline cohomology, then its mixed characteristic deformations are unobstructed; this is an analogue of the classical BTT theorem in characteristic zero. If $Z$ is ordinary, we show that it moreover admits a canonical lift to characteristic zero; this extends classical Serre-Tate theory. Our work generalises results of Achinger-Zdanowicz, Bogomolov-Tian-Todorov, Deligne-Nygaard, Ekedahl-Shepherd-Barron, Schröer, Serre-Tate, and Ward.