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On additive Deligne-Simpson problems. (English) Zbl 1442.14038

Iohara, Kenji (ed.) et al., Two algebraic byways from differential equations: Gröbner bases and quivers. Cham: Springer. Algorithms Comput. Math. 28, 271-323 (2020).
The additive Deligne-Simpson problem (ADSP) is formulated like this: Give necessary and sufficient conditions for the choice of the conjugacy classes \(C_j\in M(n,\mathbb{C})\) so that there exist irreducible tuples of matrices \(A_j\in C_j\) satisfying the condition \(A_0+\cdots +A_p=0\). The matrices \(A_j\) are interpreted as residuum-matrices of Fuchsian linear systems of ordinary differential equations on Riemann’s sphere. The author explains the ADSP and its generalization for differential equations with unramified irregular singularities.
A correspondence between spaces of solutions of these ADSPs and quiver varieties is given. As an application, the geometry of moduli spaces of meromorphic connections with unramified irregular singularities is discussed, for example, the non-emptiness of the smooth parts of moduli spaces and their connectedness.
For the entire collection see [Zbl 1444.14002].

MSC:

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
16G20 Representations of quivers and partially ordered sets
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