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We introduce the notion of regularity for a relative holonomic
-module in the sense of Monteiro Fernandes and Sabbah [Internat. Math. Res. Not. (21) (2013), 4961–4984]. We prove that the solution functor from the bounded derived category of regular relative holonomic modules to that of relative constructible complexes is essentially surjective by constructing a right quasi-inverse functor. When restricted to relative
-modules underlying a regular mixed twistor
-module, this functor satisfies the left quasi-inverse property.
Suppose we are given complex manifolds X and Y together with substacks
of modules over algebras of formal deformation
on X and
on Y, respectively. Also, suppose we are given a functor Φ from the category of open subsets of X to the category of open subsets of Y together with a functor F of prestacks from
. Then we give conditions for the existence of a canonical functor, extension of F to the category of coherent
-modules such that the cohomology associated to the action of the formal parameter
takes values in
. We give an explicit construction and prove that when the initial functor F is exact on each open subset, so is its extension. Our construction permits to extend the functors of inverse image, Fourier transform, specialisation and micro-localisation, nearby and vanishing cycles in the framework of
-modules. We also obtain the Cauchy–Kowalewskaia–Kashiwara theorem in the non-characteristic case as well as comparison theorems for regular holonomic
-modules and a coherency criterion for proper direct images of good
We prove that the k-truncated microsupport of the specialization of a complex of sheaves F along a submanifold is contained in the normal cone to the conormal bundle along the k-truncated microsupport of F. In the complex case, applying our estimates to , where is a coherent -module, we obtain new estimates for the truncated microsupport of real analytic and hyperfunction solutions. When is regular along Y we also obtain estimates for the truncated microsupport of the holomorphic solutions of the induced system along Y as well as for the nearby-cycle sheaf of when Y is a hypersurface.
A new family of sheaves has been recently studied by M. Kashiwara and P. Schapira generalizing to constructible sheaves the notion of moderate and formal cohomology. We prove comparison theorems when we regard these sheaves as solutions of a $\cal D$-module. These results are natural generalizations of those of Y. Laurent and the author.
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