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

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 Ŵ(R) 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 . We use this form to prove a lifting criterion of Grothendieck–Messing type for schemes of K3 type. The crystalline cohomology Hcrys2(XŴ(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 infinitesimal deformations of X correspond bijectively to infinitesimal deformations of the 2-display of X with its Beauville–Bogomolov form. For ordinary K3 surfaces XR we prove that the slope spectral sequence of the de Rham–Witt complex degenerates and that Hcrys2(XW(R)) has a canonical Hodge–Witt decomposition.

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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

Information

Received: 5 September 2017; Revised: 16 May 2018; Accepted: 30 September 2018; Published: 2019
First available in Project Euclid: 18 December 2018

zbMATH: 07027463
MathSciNet: MR3892249
Digital Object Identifier: 10.2140/tunis.2019.1.455

Subjects:
Primary: 14F30, 14F40

Rights: Copyright © 2019 Mathematical Sciences Publishers

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