Lifts of twisted K3 surfaces to characteristic 0

TL;DR

This paper extends Deligne's lift result from K3 surfaces to twisted K3 surfaces in positive characteristic, analyzing their deformation spaces and moduli, and applying this to derived equivalences.

Contribution

It proves that twisted K3 surfaces over positive characteristic fields can be lifted to characteristic zero, generalizing Deligne's result and exploring their deformation and moduli spaces.

Findings

Twisted K3 surfaces admit lifts to characteristic zero.

Constructs algebraic moduli spaces of twisted K3 surfaces over integers.

Shows derived equivalences are orientation preserving in positive characteristic.

Abstract

Deligne showed that every K3 surface over an algebraically closed field of positive characteristic admits a lift to characteristic 0. We show the same is true for a twisted K3 surface. To do this, we study the versal deformation spaces of twisted K3 surfaces, which are particularly interesting when the characteristic divides the order of the Brauer class. We also give an algebraic construction of certain moduli spaces of twisted K3 surfaces over $\mathrm{Spec}\mathbf{Z}$ and apply our deformation theory to study their geometry. As an application of our results, we show that every derived equivalence between twisted K3 surfaces in positive characteristic is orientation preserving.