## Duke Mathematical Journal

### Linear differential equations on the Riemann sphere and representations of quivers

Kazuki Hiroe

#### Abstract

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

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

Dates
Revised: 24 June 2016
First available in Project Euclid: 12 November 2016

https://projecteuclid.org/euclid.dmj/1478919690

Digital Object Identifier
doi:10.1215/00127094-3769640

Mathematical Reviews number (MathSciNet)
MR3626566

Zentralblatt MATH identifier
1368.16018

#### Citation

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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