Vol. 1, No. 4, 2019

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Grothendieck–Messing deformation theory for varieties of K3 type

Andreas Langer and Thomas Zink

Vol. 1 (2019), No. 4, 455–517
Abstract

Let R be an artinian local ring with perfect residue class field k. We associate to certain 2-displays over the small ring of Witt vectors Ŵ(R) a crystal on SpecR.

Let X be a scheme of K3 type over SpecR. We define a perfect bilinear form on the second crystalline cohomology group X which generalizes the Beauville–Bogomolov form for hyper-Kähler varieties over . We use this form to prove a lifting criterion of Grothendieck–Messing type for schemes of K3 type. The crystalline cohomology Hcrys2(XŴ(R)) is endowed with the structure of a 2-display such that the Beauville–Bogomolov form becomes a bilinear form in the sense of displays. If X is ordinary, the infinitesimal deformations of X correspond bijectively to infinitesimal deformations of the 2-display of X with its Beauville–Bogomolov form. For ordinary K3 surfaces XR we prove that the slope spectral sequence of the de Rham–Witt complex degenerates and that Hcrys2(XW(R)) has a canonical Hodge–Witt decomposition.

Keywords
crystalline cohomology, displays, Dieudonné 2-displays, $F$-ordinary schemes
Mathematical Subject Classification 2010
Primary: 14F30, 14F40
Milestones
Received: 5 September 2017
Revised: 16 May 2018
Accepted: 30 September 2018
Published: 14 December 2018
Authors
Andreas Langer
Department of Mathematics
University of Exeter
Devon
United Kingdom
Thomas Zink
Facultät für Mathematik
Universität Bielefeld
Bielefeld
Germany