Differentiable holonomic $AV$-modules

TL;DR

This paper investigates the structure of differentiable holonomic sheaves of $AV$-modules, revealing their local tensor product decomposition involving simple holonomic $D$-modules and finite-dimensional $gl_n$-modules.

Contribution

It establishes a local tensor product decomposition for simple differentiable holonomic sheaves of $AV$-modules, linking them to holonomic $D$-modules and $gl_n$-modules.

Findings

Simple differentiable holonomic sheaves are locally tensor products of holonomic $D$-modules and $gl_n$-modules.

When $W$ is integrable, the sheaf decomposes into a tensor product involving an associated tensor module.

The structure provides a clear classification of such sheaves in terms of well-understood modules.

Abstract

We study differentiable holonomic sheaves of $AV$-modules on a smooth quasi-projective variety. We show that a simple differentiable holonomic sheaf $M$ of $AV$-modules is locally the tensor product of a simple holonomic $D$-module and a simple finite-dimensional $gl_n$-module $W$. In particular, in the case when $W$ is integrable, $M$ is the tensor product of a simple holonomic $D$-module and the tensor module associated with $W$.