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Modules holonômes à singularités régulières et filtration de Hodge. II. (Holonomic modules at regular singularities and Hodge filtration. II). (French) Zbl 0598.14008
Astérisque 101-102, 75-117 (1983).
[For part I see Algebraic geometry, Proc. int. Conf., La Rábida/Spain 1981, Lect. Notes Math. 961, 1-21 (1982; Zbl 0523.14010).]
This is a fundamental but highly speculative paper. It contains ideas how to put a Hodge structure on intersection homology. Deligne has observed that via the Riemann-Hilbert correspondence between constructible sheaf complexes and complexes of D-modules with regular holonomic cohomology sheaves (due to Mebkhout and Kashiwara), the regular holonomic D-modules correspond to the perverse complexes, i.e. those which satisfy the axioms for the intersection complexes of Goresky and MacPherson. Start with an analytic space Y and a complex local system V on a dense open subset of the regular part of Y. The intersection complex IC(Y,V) is then isomorphic to the de Rham complex of a unique regular holonomic \(D_ X\)- module L; here \(Y\subset X\) is an embedding of Y in a smooth X. The main idea is to filter the groups IH(Y,V) by filtering L and its de Rham complex.
In the greatest generality, V is underlying a variation of mixed Hodge structure and one arrives at the notion of ”mixed holonomic D-module”. The main difficulty in the definitions seems to deal with duality of filtered D-modules. The author conjectures that his set-up works. This has partly been proved by M. Saito [Systèmes différentiels et singularités, Colloq. Luminy/France 1983, Astérisque 130, 342-351 (1985)].
For the entire collection see [Zbl 0515.00021].
Reviewer: J.H.M.Steenbrink

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32L99 Holomorphic fiber spaces
14F40 de Rham cohomology and algebraic geometry
58J10 Differential complexes
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
58A14 Hodge theory in global analysis