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

Hélène Esnault, Keiji Oguiso

Published 2014-06-11, updated 2014-08-19Version 3

We show that for a K3 surface in characteristic $p\ge 3$, there is a projective model $X_R\to {\rm Spec} \ R$ in characteristic $0$ with Picard number $1$ over a geometric generic point (Thm. 4.2). In particular, this model essentially kills all automorphisms (Thm. 5.1). We show that there is an automorphism on a supersingular K3 surface in characteristic $3$, which has positive entropy, the logarithm of a Salem number of degree $22$ (Thm. 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 $0$ (Thm. 7.5).