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Sur la théorie de Hodge des variétés algébriques à singularités isolées. (On Hodge theory of algebraic varieties with isolated singularities). (French) Zbl 0599.14007
Systèmes différentiels et singularités, Colloq. Luminy/France 1983, Astérisque 130, 272-307 (1985).
[For the entire collection see Zbl 0559.00004.]
The paper studies the mixed Hodge structures on the cohomology groups H(X), \(H_ x(X)\) and IH(X) of an algebraic variety X with isolated singularities, and of a resolution \(f : \tilde X\to X.\) The results of the paper (purity theorems of IH(X), difficult Lefschetz theorem,...) were suggested by (and are consequences of) theorems on perverse sheaves by P. Deligne, O. Gabber, A. Beilinson and I. Bernstein, viewing the relation of the above cohomology groups with the sheaves of perverse cohomology \({\mathbb{R}}f_*{\mathbb{C}}_{\tilde X}\) and the perverse sheaf \(IC_ X^{\bullet}\). The author gives ”purely transcendental” proofs of this results. He uses especially the filtred complex of de Rham \(\Omega_ X\) of X. He shows that the results on the vanishing of cohomology sheaves from F. Guillén, the author and F. Puerta: ”Théorie de Hodge via schémas cubiques” (prepublication 1972)] lead to adequate variants for the decomposition theorem of Hodge, vanishing theorem of Kodaira, difficult theorem of Lefschetz for strongly pseudo-convex varieties (in the direction of Nakano and Grauert- Riemenschneider). Further the filtered de Rham complex of a rational singularity and an isolated hypersurface singularity is studied in detail.
Reviewer: A.Brezuleanu

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F40 de Rham cohomology and algebraic geometry
14B05 Singularities in algebraic geometry
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14J17 Singularities of surfaces or higher-dimensional varieties