Abstract
On a complex manifold, the embedding of the category of regular holonomic D-modules into that of holonomic D-modules has a left quasi-inverse functor $\mathcal{M}\mapsto\mathcal{M}_{\mathrm{reg}}$, called regularization. Recall that $\mathcal{M}_{\mathrm{reg}}$ is reconstructed from the de Rham complex of $\mathcal{M}$ by the regular Riemann-Hilbert correspondence. Similarly, on a topological space, the embedding of sheaves into enhanced ind-sheaves has a left quasi-inverse functor, called here sheafification. Regularization and sheafification are intertwined by the irregular Riemann-Hilbert correspondence. Here, we study some of their properties. In particular, we provide a germ formula for the sheafification of enhanced specialization and microlocalization.
Highlights
The regular Riemann-Hilbert correspondence states that the de Rham functor induces an equivalence between the triangulated category of regular holonomic D-modules and that of C-constructible sheaves
With the aim of better understanding the rather elusive regularization functor, in this paper we study some of the properties of the ind-sheafification and sheafification functors
We provide in Appendix B a formula for the sections of a weakly constructible sheaf on a locally closed subanalytic subset, which could be of independent interest
Summary
(At the level of D-modules, the interest of such a formula is due to the lack of commutation between the de Rham functor and the restriction functor.). We provide in Appendix B a formula for the sections of a weakly constructible sheaf on a locally closed subanalytic subset, which could be of independent interest
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