Duke Mathematical Journal

Linear differential equations on the Riemann sphere and representations of quivers

Kazuki Hiroe

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Our interest in this article is a generalization of the additive Deligne–Simpson problem, which was originally defined for Fuchsian differential equations on the Riemann sphere. We extend this problem to differential equations having an arbitrary number of unramified irregular singular points, and we determine the existence of solutions of the generalized additive Deligne–Simpson problems. Moreover, we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere (namely, open embedding of the moduli spaces into quiver varieties and the nonemptiness condition of the moduli spaces). Furthermore, we detail the connectedness of the moduli spaces.

Article information

Duke Math. J., Volume 166, Number 5 (2017), 855-935.

Received: 24 December 2014
Revised: 24 June 2016
First available in Project Euclid: 12 November 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16G20: Representations of quivers and partially ordered sets
Secondary: 34M25: Formal solutions, transform techniques 34M56: Isomonodromic deformations

additive Deligne–Simpson problem linear ODE with irregular singular points middle convolution representations of quivers moduli spaces of meromorphic connections


Hiroe, Kazuki. Linear differential equations on the Riemann sphere and representations of quivers. Duke Math. J. 166 (2017), no. 5, 855--935. doi:10.1215/00127094-3769640. https://projecteuclid.org/euclid.dmj/1478919690

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