Deformations and lifts of Calabi-Yau varieties in characteristic p

Lukas Brantner (University of Oxford)

Tuesday 6th May 15:00-16:00
Maths 311B

Abstract

Homotopy theory allows us to study formal moduli problems via their tangent Lie algebras. After briefly reviewing this general paradigm, I will explain how it sheds light on deformations of Calabi-Yau varieties.  In joint work with Taelman, we prove a mixed characteristic analogue of the Bogomolov–Tian–Todorov theorem, which asserts that Calabi-Yau varieties in characteristic 0 are unobstructed. Moreover, we show that ordinary Calabi–Yau varieties in characteristic p admit canonical (and algebraisable) lifts to characteristic 0, generalising results of Serre-Tate for abelian varieties and Deligne-Nygaard for K3 surfaces. If time permits, I will conclude by discussing some intriguing questions related to our canonical lifts.

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