On a topological counterpart of regularization for holonomic 𝒟-modules

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 ℳ→ℳreg, called regularization. Recall that ℳreg is reconstructed from the de Rham complex of 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 the properties of the sheafification functor. In particular, we provide a stalk formula for the sheafification of enhanced specialization and microlocalization.