The Deligne-Simpson problem for connections on Gm with a maximally ramified 
singularity

Kulkarni, Maitreyee C.; Livesay, Neal; Matherne, Jacob P.; Nguyen, Bach; Sage, Daniel S.

doi:10.1016/j.aim.2022.108596

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The Deligne-Simpson problem for connections on G_m with a maximally ramified singularity

MPS-Authors

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Kulkarni, Maitreyee C.
Max Planck Institute for Mathematics, Max Planck Society;

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Matherne, Jacob P.
Max Planck Institute for Mathematics, Max Planck Society;

External Resource

https://doi.org/10.1016/j.aim.2022.108596
(Publisher version)

https://doi.org/10.48550/arXiv.2108.11029
(Preprint)

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Citation

Kulkarni, M. C., Livesay, N., Matherne, J. P., Nguyen, B., & Sage, D. S. (2022). The Deligne-Simpson problem for connections on G_m with a maximally ramified singularity. Advances in Mathematics, 408(Part B): 108596. doi:10.1016/j.aim.2022.108596.

Cite as: https://hdl.handle.net/21.11116/0000-000A-DB98-4

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.