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Cristaux ordinaires et coordonnées canoniques (avec la collaboration de L. Illusie). (French) Zbl 0537.14012
Surfaces algébriques, Sémin. de géométrie algébrique, Orsay 1976-78, Lect. Notes Math. 868, 80-137 (1981).
[For the entire collection see Zbl 0452.00012.]
Let $$X_ 0$$ be an ordinary K3 surface over a field of characteristic $$p\neq 2$$, so that the action of Frobenius on $$H^ 2(X,{\mathcal O})$$ is non- trivial, and let $$S$$ be the formal moduli space of deformations of $$X_ 0$$ over the local Artin W-algebras with residue field k $$(W=W(k)$$ is the ring of Witt vectors). The authors show that $$S$$ has a canonical structure of formal group and is isomorphic to $$(G_ m)_ W^{20}$$. In particular, the universal deformation $$X$$ of $$X_ 0$$ over $$S$$ defines ”canonical coordinates” $$q_ i$$ (1$$\leq i\leq 20)$$ forming a basis of the group of characters of $$S$$ such that $$S=Spf W[[q_ 1^{-1},...,q^{- 1}_{20}]].$$ The structure of formal group on $$S$$ is defined in terms of the structure of $$F$$-crystal on $$H^ 2_{DR}(X/S)$$. A similar argument enables the authors to treat the (known) case of abelian varieties. The results discussed in this report [which is a natural continuation of the author’s preceding report ibid. 58-79 (1981; Zbl 0495.14024)] give a natural extension of the Serre-Tate theory for ordinary elliptic curves to the case of K3 surfaces and abelian varieties.
Reviewer: F.L.Zak

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14L05 Formal groups, $$p$$-divisible groups 14D15 Formal methods and deformations in algebraic geometry 14F40 de Rham cohomology and algebraic geometry