Non-liftability of automorphism groups of a K3 surface in positive characteristic

TL;DR

This paper investigates the automorphism groups of K3 surfaces in positive characteristic, showing that many automorphisms do not lift to characteristic zero and constructing explicit examples with positive entropy.

Contribution

It demonstrates the non-liftability of automorphisms of K3 surfaces from positive characteristic to characteristic zero and constructs explicit automorphisms with positive entropy.

Findings

Characteristic zero models with Picard number 1 kill automorphisms

Explicit automorphism with positive entropy on supersingular K3 in characteristic 3

Existence of automorphisms with positive entropy that do not lift to characteristic zero

Abstract

We show that a characteristic $0$ model $X_R\to \Spec R$, with Picard number $1$ over a geometric generic point, of a K3 surface in characteristic $p\ge 3$, essentially kills all automorphisms (Theorem 5.1). We show that there is an explicitely constructed automorphism on a supersingular K3 surface in characteristic $3$, which has positive entropy, the logarithm of a Salem number of degree $22$ (Theorem 6.4). In particular it does not lift to characteristic $0$. In addition, we show that in any large characteristic, there is an automorphism of a supersingular K3 which has positive entropy and does not lift to characteristic $0$ (Theorem 7.5).