Differentiable holonomic $AV$-modules

TL;DR

This work generalizes the theory of $\mathcal{D}$-modules to differentiable $AV$-modules by working with the topologically completed sheaf $\widehat{\mathcal{AV}}$ on a smooth quasi-projective variety. It develops a robust framework based on Gelfand–Kirillov dimension to define holonomicity, proves finite length for holonomic differentiable modules, and establishes a local tensor-product structure: every simple holonomic $\widehat{\mathcal{AV}}$-module is locally of the form $T(P,W)$ with $P$ a simple holonomic $\mathcal{D}$-module and $W$ a simple finite-dimensional $\mathfrak{gl}_n$-module. When $W$ is integrable, simple holonomic modules globally decompose as $\mathcal{P} \otimes \mathcal{J}^W$; for non-integrable $W$ a charged $\mathcal{D}$-module description arises. The paper also characterizes $\mathcal{O}$-coherent $\widehat{\mathcal{AV}}$-modules as vector bundles and studies the role of central characters in gl_n, including Rudakov and gauge-module constructions, which illuminate how simple differentiable holonomic modules glue across charts and their relations to tensor 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 $W$.

Theorems & Definitions (60)

• Lemma 1.1
• proof
• Lemma 1.2
• Lemma 1.3
• Definition 2.1
• Theorem 2.2: BI23
• Definition 2.3
• Lemma 2.4
• Proposition 2.5
• proof