Graded Holonomic D-modules on Monomial Curves

TL;DR

This paper classifies simple graded holonomic D-modules on affine monomial curves, extending the classification to the first Weyl algebra, and analyzes their extensions in the graded setting.

Contribution

It provides a classification of simple graded holonomic D-modules on monomial curves and computes their extensions, including the case of the first Weyl algebra.

Findings

Classified simple graded D-modules on affine monomial curves

Computed extensions of these modules

Extended classification results to the first Weyl algebra

Abstract

In this paper, we study the holonomic $D$-modules when $D$ is the ring of $k$-linear differential operators on $A = k[\Gamma]$, the coordinate ring of an affine monomial curve over the complex numbers $k = \mathbb C$. In particular, we consider the graded case, and classify the simple graded $D$-modules and compute their extensions. The classification over the first Weyl algebra $D = A_1(k)$ is obtained as a special case.