Construction of quasi-canonical liftings of K3 surfaces of finite height in odd characteristic

TL;DR

This paper constructs a quasi-canonical lifting of finite height K3 surfaces over finite fields of odd characteristic, extending previous results to include characteristic 3, using display-theoretic deformation theory.

Contribution

It introduces a new construction of quasi-canonical liftings for K3 surfaces in odd characteristic, utilizing display theory and crystalline cohomology analysis.

Findings

Constructed quasi-canonical liftings for K3 surfaces in characteristic p≥3.

Extended previous results from p≥5 to p=3.

Analyzed display structures of crystalline cohomology in deformations.

Abstract

We construct a quasi-canonical lifting of a $K3$ surface of finite height over a finite field of characteristic $p\geq3$. Such results are previously obtained by Nygaard-Ogus when $p\geq5$. For this purpose, we use the display-theoretic deformation theory developed by Langer, Zink, and Lau. We study the display structure of the crystalline cohomology of deformations of a $K3$ surface of finite height in terms of the Dieudonn\'e display of the enlarged formal Brauer group.