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Hodge structure via filtered \({\mathcal D}\)-modules. (English) Zbl 0621.14008
Systèmes différentiels et singularités, Colloq. Luminy/France 1983, Astérisque 130, 342-351 (1985).
[For the entire collection see Zbl 0559.00004.]
This is an announcement of fundamental work which is still unpublished at this time of writing. It concerns the author’s program to develop a transcendental analogue of Deligne’s theory of \(\ell\)-adic sheaves [P. Deligne, Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014)]. The main result is the definition of the category of polarizable Hodge modules on a complex manifold X. A polarizable Hodge module has two aspects: a regular holonomic \({\mathcal D}_ X\)-module \({\mathcal M}\) with a good filtration F and a perverse sheaf K of \({\mathbb{Q}}\)-vector spaces. These are connected via the Riemann-Hilbert correspondence: there is given an isomorphism between DR(\({\mathcal M})\) and \(K\otimes {\mathbb{C}}\) in the derived category. The idea to use this formalism for Hodge theory goes back to J.-L. Brylinski [Astérisque 101/102, 75-117 (1983; Zbl 0598.14008)]. These data have to satisfy lots of conditions. The category of Hodge modules on X is defined by induction on dim(X). The induction step uses the formalism of vanishing cycles for perverse sheaves (Deligne) and \({\mathcal D}\)-modules (Malgrange-Kashiwara).
Applications of this theory include a Hodge-theoretic proof of the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber and a pure Hodge structure on the intersection homology groups of algebraic varieties. Details have appeared in two recent RIMS reports of the author.
Reviewer: J.H.M.Steenbrink

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F40 de Rham cohomology and algebraic geometry
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)