The Deligne-Simpson problem for connections on $\mathbb{G}_m$ with a maximally ramified singularity

TL;DR

This paper extends the classical Deligne-Simpson problem to include maximally ramified singularities on connections over rac{G}_m, providing solutions and characterizations for rigid cases with unipotent monodromy.

Contribution

It formulates and solves a generalized Deligne-Simpson problem incorporating toral singularities and maximally ramified points on rac{G}_m, advancing the understanding of irregular singularities.

Findings

Solved the Deligne-Simpson problem for connections with maximally ramified singularities.

Characterized all such connections that are rigid with unipotent monodromy.

Extended the theory to include toral singularities with regular semisimple leading terms.

Abstract

The classical additive Deligne-Simpson problem is the existence problem for Fuchsian connections with residues at the singular points in specified adjoint orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem in terms of quiver varieties. A more general version of this problem, solved by Hiroe, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne-Simpson problem in which certain ramified singularities are allowed. These allowed singular points are called toral singularities; they are singularities whose leading term with respect to a lattice chain filtration is regular semisimple. We solve this problem in the important special case of connections on $\mathbb{G}_m$ with a maximally ramified singularity at $0$ and possibly an additional regular singular point at infinity. We also give a complete characterization of all such connections which are rigid, under the additional hypothesis of unipotent monodromy at infinity.