Regularization of Relative Holonomic $\mathcal{D}$-Modules

Abstract

Let $X$ and $S$ be complex analytic manifolds where $S$ plays the role of a parameter space. Using the sheaf ${\mathcal{D}}_{X\times S/S}^{\infty }$ of relative differential operators of infinite order, we construct functorially the regular holonomic ${\mathcal{D}}_{X\times S/S}$-module ${\mathcal{M}}_{\mathrm{reg}}$ associated to a relative holonomic ${\mathcal{D}}_{X\times S/S}$-module $\mathcal{M}$, extending to the relative case classical theorems by Kashiwara–Kawai: denoting by ${\mathcal{M}}^{\infty }$ the tensor product of $\mathcal{M}$ by ${\mathcal{D}}_{X\times S/S}^{\infty }$ we make ${\mathcal{M}}^{\infty }$ explicit in terms of the sheaf of holomorphic solutions of $\mathcal{M}$. As a consequence of the relative Riemann–Hilbert correspondence we conclude that ${\mathcal{M}}^{\infty }$ and ${\mathcal{M}}_{\mathrm{reg}}^{\infty }$ are isomorphic.