Un scindage de la filtration de Hodge pour certaines variétés algébriques sur les corps locaux. (A splitting of the Hodge filtration for certain algebraic varieties over local fields). (French) Zbl 0599.14018

Wintenberger, Jean-Pierre

Ann. Math. (2) 119, 511-548 (1984).

Let K be an unramified local field of characteristic zero with perfect residue class field of characteristic p. The de Rham cohomology \(H^ m_{DR}({\mathcal X})\) of a \(variety\quad {\mathcal X}\) smooth and proper over K has a filtration by degrees of differential forms. When \({\mathcal X}\) has good reduction the Frobenius lifts to \(H_{DR}\) as a semi-linear mapping over K. J.-M. Fontaine [Ann. Math., II. Ser. 115, 529-577 (1982; Zbl 0544.14016)] has defined ”weak admissibility” for such structures and verified that it holds for \({\mathcal X}\) abelian, or for certain cohomological conditions on extensions of \({\mathcal X}\) over the ring of integers of K.
The present article shows that when the condition of weak admissibility holds the filtration of the de Rham cohomology splits as in the ordinary case. The structures obtained are compatible with \(\otimes\) and this allows a Tannakian definition of certain algebraic groups H, \(H^{e\ell}\). In particular H has also a description of Mumford-Tate type. Finally the author introduces a new polygon for these structures (as well as Newton, Hodge).

Reviewer: G.Horrocks

MSC:

14F40	de Rham cohomology and algebraic geometry
14F30	\(p\)-adic cohomology, crystalline cohomology
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)

Keywords:

Mumford-Tate group; weak admissibility; filtration of the de Rham cohomology splits

Citations:

Zbl 0544.14016