Grothendieck-Messing deformation theory for varieties of K3-type

Abstract

Let R be an artinian local ring with perfect residue class eld 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 de ne a perfect bilinear form on the second crystalline cohomology group X which generalizes the Beauville-Bogomolov form for hyper-K ahler varieties over C. We use this form to prove a lifting criterion of Grothendieck- Messing type for schemes of K3-type. The crystalline cohomology H2 crys(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 in nitesimal deformations of X correspond bijectively to in nitesimal 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 H2 crys(X=W(R)) has a canonical Hodge- Witt decomposition.