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Induced \({\mathcal D}\)-modules and differential complexes. (English) Zbl 0705.32005
Let \(X\to Y\) be a proper morphism of complex manifolds or of smooth algebraic varieties. It is known that one can define for any bounded complex \(M^*\) of \({\mathcal D}_ X\)-modules with coherent cohomologies a duality isomorphism \(f_*{\mathbb{D}}M^*\to {\mathbb{D}}f_*M^*.\) In this formula \(f_*\) is the direct image of \({\mathcal D}\)-modules and \({\mathbb{D}}\) is the dual functor \({\mathbb{R}} Hom_{{\mathcal D}_ X}(M^*,{\mathcal D}_ X(d_ x))\) (case of right \({\mathcal D}_ X\)-modules).
The aim of this paper is, in particular, to give a new proof of this result, based on the notion of induced \({\mathcal D}_ X\)-module introduced by the author. These are the \({\mathcal D}_ X\)-modules of the type \(L\otimes_{{\mathcal O}_ X}{\mathcal D}_ X\), for some \({\mathcal O}_ X\)- module L. Applying the functor \(\otimes_{{\mathcal D}_ X}{\mathcal O}_ X\) one gets the notion of differential morphisms \(L\to L'\) which are \({\mathcal D}_ X\) linear map induced by \({\mathcal D}_ X\)-linear morphisms \(L\otimes {\mathcal D}_ X\to L'\otimes {\mathcal D}_ X\). The author introduces also the notion of differential morphisms of finite order p which correspond to maps \(L\to L'\otimes F_ p{\mathcal D}_ X\). In § 1, he proves some equivalences of categories quite useful for the next sections:
The category \(D^ b({\mathcal O}_ X,Diff)\) (resp. \(D^ b({\mathcal O}_ X,Diff)^ f)\) of complex of \({\mathcal O}_ X\)-modules (resp. with finite order differential), is equivalent to \(D^ b({\mathcal O}_ X)\). In the first derived category are inverted the D-quasi-isomorphism, i.e. those morphism of complexes which via \(\otimes_{{\mathcal O}_ X}{\mathcal D}_ X\) induce quasi-isomorphisms in \(C^ b({\mathcal D}_ X)\). Similarly the author defines \(D^ b({\mathcal O}_ X,Diff)_{coh}\) via this equivalence. In § 2 and § 3, he obtains then his main results: First, definition of dual functors \(in\) both categories and then proof of the compatibility of \({\mathbb{D}}\), with the De Rham functor in the holonomic case, and with \(f^*\) in general. Finally in the last § 4, the author introduces the concept of diagonal pairing and gives two applications of it, to a definition of the dual functor \({\mathbb{D}}\) first and then to a simplification in the proof of the Riemann Hilbert correspondence.
Reviewer: J.M.Granger

MSC:
32C37 Duality theorems for analytic spaces
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14F40 de Rham cohomology and algebraic geometry
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