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

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.

There is a mistake in the way Theorem 3.3 is formulated, leading to a gap in the proof, as it currently assumes implicitly connectedness of some moduli spaces