Picard groups of differential operators on curves

TL;DR

This paper proves that the exact sequence describing the Picard group of differential operators on complex affine curves (excluding the affine line) is split, clarifying the structure of autoequivalences of D(X)-modules.

Contribution

It demonstrates that the exact sequence for Pic(D) on complex affine curves is split, providing new structural insight into autoequivalences of D(X)-modules.

Findings

The exact sequence for Pic(D) is split.

Clarifies the structure of autoequivalences on D(X)-modules.

Enhances understanding of differential operators on curves.

Abstract

Let X be a complex affine curve (not isomorphic to the affine line), and let Pic(D) be the group of autoequivalences of the category of D(X)-modules. Cannings and Holland have shown that Pic(D) fits into an exact sequence in which the other groups can be considered known: we show that this sequence is split.