Tempered solutions of $\mathcal D$-modules on complex curves and formal invariants

TL;DR

This paper establishes a fully faithful functor from formal holonomic $$-modules on complex curves to subanalytic sheaves of tempered solutions, linking formal invariants with analytic solutions.

Contribution

It introduces a new functor connecting formal holonomic modules with tempered solutions, enhancing understanding of their invariants and solution sheaves.

Findings

The subanalytic sheaf of tempered solutions induces a fully faithful functor.

Results linking tempered solutions with classical formal and analytic invariants.

New insights into the structure of holonomic $$-modules on complex curves.

Abstract

Let $X$ be a complex analytic curve. In this paper we prove that the subanalytic sheaf of tempered holomorphic solutions of $\mathcal D_X$-modules induces a fully faithful functor on a subcategory of germs of formal holonomic $\mathcal D_X$-modules. Further, given a germ $\mathcal M$ of holonomic $\mathcal D_X$-module, we obtain some results linking the subanalytic sheaf of tempered solutions of $\mathcal M$ and the classical formal and analytic invariants of $\mathcal M$.