#### Vol. 1, No. 4, 2019

 Recent Issues Volume 3, Issue 1 Volume 2, Issue 4 Volume 2, Issue 3 Volume 2, Issue 2 Volume 2, Issue 1 Volume 1, Issue 4 Volume 1, Issue 3 Volume 1, Issue 2 Volume 1, Issue 1
 The Journal About the Journal Editorial Board Subscriptions Submission Guidelines Submission Form Ethics Statement Editorial Login ISSN (electronic): 2576-7666 ISSN (print): 2576-7658 Author Index To Appear Other MSP Journals
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 $\stackrel{̂}{W}\left(R\right)$ 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 ${H}_{crys}^{2}\left(X∕\stackrel{̂}{W}\left(R\right)\right)$ 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.

##### Keywords
crystalline cohomology, displays, Dieudonné 2-displays, $F$-ordinary schemes
##### Mathematical Subject Classification 2010
Primary: 14F30, 14F40