Abstract
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.
Citation
Andreas Langer. Thomas Zink. "Grothendieck–Messing deformation theory for varieties of K3 type." Tunisian J. Math. 1 (4) 455 - 517, 2019. https://doi.org/10.2140/tunis.2019.1.455
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