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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

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32C38 Sheaves of differential operators and their modules, $$D$$-modules