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Regular holonomic D-modules and distributions on complex manifolds. (English) Zbl 0613.14015
Complex analytic singularities, Proc. Semin., Ibaraki/Jap. 1984, Adv. Stud. Pure Math. 8, 199-206 (1987).
[For the entire collection see Zbl 0607.00005.]
This paper deals with the construction of the complex conjugate functor in the category \(RH({\mathcal D}_ X)\) of regular holonomic \({\mathcal D}_ X\)-modules on a complex manifold X. Namely, let \((X,{\mathcal O}_ X)\) be a complex manifold and let \({\mathcal D}_ X\) be the sheaf of differential operators on X. The de Rham functor \({\mathcal D}{\mathcal R}_ X=R{\mathcal H}om_{{\mathcal D}_ X}({\mathcal O}_ X,*)\) gives an equivalence of \(RH({\mathcal D}_ X)\) and the category \(Perv({\mathbb{C}}_ X)\) of perverse sheaves of \({\mathbb{C}}\)-vector spaces on X. To a perverse sheaf \({\mathcal F}^.\) we can associate its complex conjugate \(\bar {\mathcal F}^.\). Then it is easily checked that \(\bar {\mathcal F}^.\) is also perverse. The problem here is how to construct the corresponding functor \(c: RH({\mathcal D}_ X)\to RH({\mathcal D}_ X)\) given by \(\overline{{\mathcal D}{\mathcal R}_ X({\mathfrak M})}={\mathcal D}{\mathcal R}_ X({\mathfrak M}^{c})\). The solution to this problem is given as follows. Let \(\bar X\) be the complex conjugate of X and \(\bar{\mathfrak M}\) the complex conjugate of \({\mathfrak M}\). Denoting by \({\mathcal D}b_{X_{{\mathbb{R}}}}\) the sheaf of distributions on the underlying real manifold \(X_{{\mathbb{R}}}\) of X, \({\mathfrak M}^{c}\) is given by \({\mathcal T}or_ n^{{\mathcal D}_ X}(\Omega^ n_{\bar X} \otimes_{{\mathcal O}_{\bar X}} {\mathcal D}b_{X_{{\mathbb{R}}}},\bar {\mathfrak M})\) where \(n=\dim(X)\) and \(\Omega^ n_{\bar X}\) denotes the sheaf of the highest degree differential forms on \(\bar X\).
Reviewer: M.Muro

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C38 Sheaves of differential operators and their modules, \(D\)-modules