Small Resolutions and Non-Liftable Calabi-Yau threefolds

TL;DR

This paper constructs examples of smooth Calabi-Yau threefolds over finite fields that cannot be lifted to characteristic zero, using properties of small resolutions of ordinary double points.

Contribution

It introduces new methods to produce non-liftable Calabi-Yau threefolds via small resolutions, expanding understanding of their behavior in positive characteristic.

Findings

Constructed a Calabi-Yau threefold over _3 that does not lift to characteristic zero.

Produced a Calabi-Yau threefold over _5 with obstructed deformation.

Generated numerous examples of non-liftable Calabi-Yau algebraic spaces over _p.

Abstract

We use properties of small resolutions of the ordinary double point in dimension three to construct smooth non-liftable Calabi-Yau threefolds. In particular, we construct a smooth projective Calabi-Yau threefold over $\F_3$ that does not lift to characteristic zero and a smooth projective Calabi-Yau threefold over $\F_5$ having an obstructed deformation. We also construct many examples of smooth Calabi-Yau algebraic spaces over $\F_p$ that do not lift to algebraic spaces in characteristic zero.