Relèvements modulo \(p^ 2\) et décomposition du complexe de de Rham. (Lifting modulo \(p^ 2\) and decomposition of the de Rham complex). (French) Zbl 0632.14017

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 result to the Kodaira-Akizuki-Nakano vanishing theorem hence giving an elementary proof of that result and finally a generalisation to the corresponding degeneration for cohomology of forms with logarithmic poles (along a normal crossing divisor).
Reviewer: T.Ekedahl


14F40 de Rham cohomology and algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32J15 Compact complex surfaces
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