# Index of rigidity of differential equations and Euler characteristic of their spectral curves

## TL;DR

This paper establishes a connection between the index of rigidity of irregular differential equations on Riemann surfaces and the Euler characteristic of their spectral curves, linking local invariants through Milnor formulas.

## Contribution

It demonstrates a novel coincidence between the index of rigidity and the Euler characteristic of spectral curves, and compares local invariants via Milnor formulas.

## Key findings

- Index of rigidity equals Euler characteristic of spectral curves.
- Milnor formula links irregularity and Milnor number.
- Spectral curves are called irregular spectral curves.

## Abstract

We show a coincidence of index of rigidity of differential equations with irregular singularities on a compact Riemann surface and Euler characteristic of the associated spectral curves which are recently called irregular spectral curves. Also we present a comparison of local invariants, so called Milnor formula which links the Komatsu-Malgrange irregularity of differential equations and Milnor number of the spectral curves.