Grothendieck–Messing deformation theory for varieties of K3 type

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 W(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-Kahler varieties over ℂ. We use this form to prove a lifting criterion of Grothendieck–Messing type for schemes of K3 type. The crystalline cohomology Hcrys2(X∕W(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 X∕R we prove that the slope spectral sequence of the de Rham–Witt complex degenerates and that Hcrys2(X∕W(R)) has a canonical Hodge–Witt decomposition.