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Filtered perverse complexes. (English) Zbl 0923.14011

Summary: We introduce the notion of filtered perversity of a filtered differential complex on a complex analytic manifold \(X\), without any assumptions of coherence, with the purpose of studying the connection between the pure Hodge modules and the \(L^2\)-complexes. We show that if a filtered differential complex \(({\mathcal M}^\bullet, F_\bullet)\) is filtered perverse then \(DR^{-1}({\mathcal M}^\bullet,F_\bullet)\) is isomorphic to a filtered \({\mathcal D}\)-module; a coherence assumption on the cohomology of \(({\mathcal M}^\bullet, F_\bullet)\) implies that, in addition, this \({\mathcal D}\)-module is holonomic.
We show the converse: The de Rham complex of a holonomic Cohen-Macaulay filtered \({\mathcal D}\)-module is filtered perverse.

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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