zbMATH — the first resource for mathematics

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

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