The Deligne-Simpson problem via 2-Calabi-Yau categories

TL;DR

This paper offers a concise proof of a key condition in the Deligne-Simpson problem using advanced category theory and local neighborhood theorems, enhancing understanding of monodromy representations.

Contribution

It presents a new proof of Crawley-Boevey's necessary condition leveraging 2-Calabi-Yau categories and local neighborhood theorems, connecting algebraic geometry and category theory.

Findings

Validated the necessity of Crawley-Boevey's condition

Connected 2-Calabi-Yau categories with monodromy problems

Provided a simplified proof approach for the Deligne-Simpson problem

Abstract

We provide a short proof of the necessity of Crawley-Boevey's condition in his solution to the Deligne-Simpson problem. The proof relies on the local neighbourhood theorem for $2$-Calabi-Yau categories due to Davison together with Crawley-Boevey's sufficient condition for the existence of local systems with prescribed conjugacy classes of monodromy around the punctures.