Calabi-Yau threefolds in positive characteristic

TL;DR

This paper reviews Calabi-Yau threefolds in positive characteristic, highlighting their unique deformation properties and the fact that unlike lower dimensions, some cannot be lifted to characteristic zero, making them intriguing in algebraic geometry.

Contribution

It provides an overview of the deformation and liftability properties of Calabi-Yau threefolds in positive characteristic, emphasizing their differences from lower-dimensional cases.

Findings

Calabi-Yau threefolds in positive characteristic can be non-liftable to characteristic zero.

Elliptic curves and K3 surfaces are all liftable, unlike some threefolds.

Calabi-Yau threefolds in positive characteristic are still poorly understood and are a subject of ongoing research.

Abstract

In this note, an overview of Calabi-Yau varieties in positive characteristic is presented. Although Calabi-Yau varieties in characteristic zero are unobstructed, there are examples of Calabi-Yau threefolds in positive characteristic which cannot be lifted to characteristic zero, although one-dimensional and two-dimensional Calabi-Yau varieties, i.e., elliptic curves and K3 surfaces, are all liftable to characteristic zero. In this respect, Calabi-Yau threefolds in positive characteristic are interesting in view of deformation theory and they are still very mysterious.