Regularization of relative holonomic D-modules

TL;DR

This paper extends classical theorems on regular holonomic D-modules to the relative setting over complex manifolds, constructing a functorial regularization process using infinite order differential operators.

Contribution

It introduces a functorial construction of the regular holonomic D-module in the relative case, generalizing Kashiwara-Kawai theorems to parameterized families.

Findings

Explicit description of the tensor product with the sheaf of infinite order differential operators.

Proof of isomorphism between the tensor products of the original and regularized modules.

Extension of classical regularity results to the relative setting over complex manifolds.

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.