Linear differential equations on the Riemann sphere and representations of quivers

Abstract

Our interest in this paper is a generalization of the additive Deligne-Simpson problem which is originally defined for Fuchsian differential equations on the Riemann sphere. We shall extend this problem to differential equations having an arbitrary number of unramified irregular singular points and 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 is given and the non-emptiness condition of the moduli spaces is determined. Furthermore the connectedness of the moduli spaces is shown.

62 pages, some errors and typos are corrected

References in corpus (12)

• Symplectic Manifolds and Isomonodromic Deformations
• Geometry of Multiplicative Preprojective Algebra
• Irregular connections and Kac-Moody root systems
• Moduli of unramified irregular singular parabolic connections on a smooth projective curve
• Twisted wild character varieties
• Rigid irregular connections on P^1
• Middle Convolution and Harnad Duality
• Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations
• Moduli spaces of irregular singular connections
• Moduli space of irregular singular parabolic connections of generic ramified type on a smooth projective curve
• Linear differential equations on ${\mathbb{P}}^1$ and root systems
• Introduction to middle convolution for differential equations with irregular singularities

Cited by in corpus (7)

• Diagrams for nonabelian Hodge spaces on the affine line
• The moduli stack of parabolic bundles over the projective line, quiver representations, and the Deligne-Simpson problem
• The Deligne-Simpson problem for connections on ${\mathbb{G}}_m$ with a maximally ramified singularity
• Moduli spaces of meromorphic connections and quiver varieties
• Lecture Notes on Quiver Representations and Moduli Problems in Algebraic Geometry
• Diagrams and irregular connections on the Riemann sphere
• Moduli spaces of meromorphic connections, quiver varieties, and integrable deformations