Relèvements modulo \(p^ 2\) et décomposition du complexe de de Rham. (Lifting modulo \(p^ 2\) and decomposition of the de Rham complex)
type: Item
instance of: scholarly article
title: Relèvements modulo \(p^ 2\) et décomposition du complexe de de Rham. (Lifting modulo \(p^ 2\) and decomposition of the de Rham complex)
zbMATH Open document ID: 0632.14017
DOI: 10.1007/BF01389078
author: Pierre Deligne
author: Luc Illusie
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143480
review text: This paper presents what is certainly the most elementary and quite possibly the simplest proof of the degeneration of the Hodge to de Rham spectral sequence for a Moishezon manifold which has been obtained so far. The proof proceeds by reduction to the case of positive characteristic \(p\) with the extra assumption that it lifts to modulo \(p^ 2\). Under this assumption and the additional assumption that the dimension of the variety is less than p the authors show that the de Rham complex considered as a complex of sheaves on the variety is quasi- isomorphic to a complex with trivial differential thence the degeneration of the ``cohomology of cohomology sheaves'' to de Rham spectral sequence. As the \(E_ 2\)-term of this spectral sequence is known to have the same dimension as the Hodge cohomology the result follows. This decomposition is obtained by trying to construct a quasi-isomorphism from the cohomology of the de Rham complex to the complex itself. Locally, where one may assume that one has a lifting of the Frobenius map this can be done in a canonical way (depending on the lifting of the Frobenius map). Then the dependence on the lifting of the Frobenius map is analysed and one sees that the maps associated to two choices of lifting can be connected by an explicit homotopy. This last result (and the subsequent verification of a transitivity condition) is what allows one to glue everything together globally. It should be noted that this kind of construction has been with us since the beginning of the study of the de Rham complex in positive characteristic but usually one has made a choice of both a lifting of the variety and the Frobenius and then one has to settle for much weaker results which do not suffice to show degeneration (and indeed it is false if one does not assume a lifting of the variety). The article then continues with a study of the dependence of the decomposition on the lifting, a study of what happens in a family, an application of the resul…
reviewed by: Torsten Ekedahl
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 32J15
zbMATH DE Number: 4027650
zbMATH Keywords: Hodge decomposition
zbMATH Keywords: Moishezon manifold
zbMATH Keywords: lifting of the Frobenius map
zbMATH Keywords: Kodaira-Akizuki-Nakano vanishing theorem
Wikidata QID: Q29027198
MaRDI profile type: Publication
cites work: Note on Kodaira-Spencer's proof of Lefschetz theorems
cites work: Q3213240
cites work: Notes on Crystalline Cohomology. (MN-21)
cites work: Q3241273
cites work: Théoreme de Lefschetz et critères de dégénérescence de suites spectrales
cites work: Théorie de Hodge. II. (Hodge theory. II)
cites work: Logarithmic de Rham complexes and vanishing theorems
cites work: Complexe de de\thinspace Rham-Witt et cohomologie cristalline
cites work: Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin
cites work: Algebraic solutions of differential equations (p-curvature and the Hodge filtration)
cites work: Algebraic spaces
cites work: On a Differential-Geometric Method in the Theory of Analytic Stacks
cites work: Quasi-elliptic surfaces in characteristic three
cites work: Frobenius and the Hodge filtration (estimates)
cites work: Pathologies of Modular Algebraic Surfaces
cites work: Q4402617
description: scientific article; zbMATH DE number 4027650