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Abstract
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Let
be an artinian local ring with perfect residue class field
. We associate to certain
-displays over the small
ring of Witt vectors
a crystal on
.
Let
be a scheme
of K3 type over
.
We define a perfect bilinear form on the second crystalline cohomology group
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
is endowed with the
structure of a
-display
such that the Beauville–Bogomolov form becomes a bilinear form in the sense of displays. If
is ordinary, the infinitesimal
deformations of
correspond bijectively to infinitesimal deformations of the
-display
of
with its Beauville–Bogomolov form. For ordinary K3 surfaces
we
prove that the slope spectral sequence of the de Rham–Witt complex degenerates and
that
has a canonical Hodge–Witt decomposition.
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Keywords
crystalline cohomology, displays, Dieudonné 2-displays,
$F$-ordinary schemes
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Mathematical Subject Classification 2010
Primary: 14F30, 14F40
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Milestones
Received: 5 September 2017
Revised: 16 May 2018
Accepted: 30 September 2018
Published: 14 December 2018
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