Deformations and Lifts of Calabi-Yau Varieties in Characteristic $p$

TL;DR

This paper investigates the deformation theory of Calabi-Yau varieties in characteristic p, establishing unobstructedness results and canonical lifts to characteristic zero using derived algebraic geometry techniques.

Contribution

It proves a mixed characteristic analogue of the Bogomolov-Tian-Todorov theorem and extends the Serre-Tate theorem to ordinary Calabi-Yau varieties.

Findings

Calabi-Yau varieties in characteristic p are unobstructed under certain conditions.

Ordinary Calabi-Yau varieties admit canonical lifts to characteristic zero.

The work uses derived algebraic geometry to achieve these results.

Abstract

We study deformations of Calabi-Yau varieties in characteristic $p$ using techniques from derived algebraic geometry. We prove a mixed characteristic analogue of the Bogomolov-Tian-Todorov theorem (which states that Calabi-Yau varieties in characteristic $0$ are unobstructed), and we show that ordinary Calabi-Yau varieties admit canonical lifts to characteristic $0$, generalising the Serre-Tate theorem on ordinary abelian varieties.