Regularization of relative holonomic D-modules

Abstract

Let $X$ and $S$ be complex analytic manifolds where $S$ plays the role of a parameter space. Using the sheaf $\DXS^{\infty}$ of relative differential operators of infinite order, we construct functorially the regular holonomic $\DXS$-module $\shm_{reg}$ associated to a relative holonomic $\DXS$-module $\shm$, extending to the relative case classical theorems by Kashiwara-Kawai: denoting by $\shm^{\infty}$ the tensor product of $\shm$ by $\DXS^{\infty}$ we explicit $\shm^{\infty}$ in terms of the sheaf of holomorphic solutions of $\shm$ and prove that $\shm^{\infty}$ and $\shm_{reg}^{\infty}$ are isomorphic.