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 $\hat{W}\left(R\right)$ 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 $ℂ$. We use this form to prove a lifting criterion of Grothendieck–Messing type for schemes of K3 type. The crystalline cohomology $H_{crys}^2\left(X∕\hat{W}\left(R\right)\right)$ 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 $H_{crys}^2\left(X∕W\left(R\right)\right)$ has a canonical Hodge–Witt decomposition.