Local Fourier transform and blowing up

TL;DR

This paper studies the resolution of ramified irregular singularities in meromorphic connections using local Fourier transforms, linking singularity invariants to plane curve geometry and knot theory.

Contribution

It provides a criterion for resolving ramified singularities of connections and introduces an analogue of Puiseux characteristics for connections.

Findings

Characterization of when a connection's singularity can be resolved

Relation between irregularity and plane curve invariants

Definition of a Puiseux characteristic analogue for connections

Abstract

We consider a resolution of ramified irregular singularities of meromorphic connections on a formal disk via local Fourier transforms. A necessary and sufficient condition for an irreducible connection to have a resolution of the ramified singularity is determined as an analogy of the blowing up of plane curve singularities. We also relate the irregularity of Komatsu and Malgrange of connections to the intersection numbers and the Milnor numbers of plane curve germs. Finally, we shall define an analogue of Puiseux characteristics for connections and find an invariant of the family of connections with the fixed Puiseux characteristic by means of the structure of iterated torus knots of the plane curve germs.