Lifts of twisted K3 surfaces to characteristic 0

Abstract

Deligne [6] 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 use this study to analyze the geometry of certain moduli spaces of twisted K3 surfaces in mixed and positive characteristic. As an application of our results, we show that every derived equivalence between twisted K3 surfaces in positive characteristic is orientation preserving.

Derived isogenies and isogenies for abelian surfaces

“…In [4], Bragg has shown that a twisted K3 surface can be lifted to characteristic 0. Though his method can not be directly applied to twisted abelian surfaces, one can still obtain a lifting result for twisted abelian surfaces via using the Kummer construction.…”

“…Proof. The existence of such lift is ensured by [4,Theorem 7.3], [25, Lemma 3.9] and Proposition 2.2.1. Roughly speaking, let S → Km(X) be the associated twisted Kummer surface via the isomorphism (2.2.3).…”

“…Roughly speaking, let S → Km(X) be the associated twisted Kummer surface via the isomorphism (2.2.3). Then [4,Theorem 7.3] asserts that there exists a lifting S → S of S → Km(X) such that the specialization map of Néron-Severi groups is an isomorphism…”