Duke Mathematical Journal

Linear differential equations on the Riemann sphere and representations of quivers

Kazuki Hiroe

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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1478919690

Digital Object Identifier
doi:10.1215/00127094-3769640

Mathematical Reviews number (MathSciNet)
MR3626566

Zentralblatt MATH identifier
1368.16018

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

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

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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References

  • [1] D. Arinkin, Rigid irregular connections on $\mathbb{P}^{1}$, Compos. Math. 146 (2010), 1323–1338.
  • [2] W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Universitext, Springer, New York, 2000.
  • [3] P. Boalch, Symplectic manifolds and isomonodromic deformations, Adv. Math. 163 (2001), 137–205.
  • [4] P. Boalch, Simply-laced isomonodromy systems, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 1–68.
  • [5] P. Boalch, Geometry and braiding of Stokes data: Fission and wild character varieties, Ann. of Math. (2) 179 (2014), 301–365.
  • [6] P. Boalch, Irregular connections and Kac-Moody root systems, preprint, arXiv:0806.1050v1 [math.DG].
  • [7] P. Boalch and D. Yamakawa, Twisted wild character varieties, preprint, arXiv:1512.08091v1 [math.AG].
  • [8] C. L. Bremer and D. S. Sage, Moduli spaces of irregular singular connections, Int. Math. Res. Not. IMRN 2013, no. 8, 1800–1872.
  • [9] W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compos. Math. 126 (2001), 257–293.
  • [10] W. Crawley-Boevey, On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero, Duke Math. J. 118 (2003), 339–352.
  • [11] W. Crawley-Boevey and M. P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), 605–635.
  • [12] W. Crawley-Boevey and P. Shaw, Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem, Adv. Math. 201 (2006), 180–208.
  • [13] M. Dettweiler and S. Reiter, An algorithm of Katz and its application to the inverse Galois problem, J. Symbolic Comput. 30 (2000), 761–798.
  • [14] M. Dettweiler and S. Reiter, Middle convolution of Fuchsian systems and the construction of rigid differential systems, J. Algebra 318 (2007), 1–24.
  • [15] V. Ginzburg, “Lectures on Nakajima’s quiver varieties” in Geometric Methods in Representation Theory, I (Grenoble, 2008), Sémin. Congr. 24-I, Soc. Math. France, Paris, 2012, 145–219.
  • [16] K. Hiroe, Linear differential equations on $\mathbb{P}^{1}$ and root systems, J. Algebra 382 (2013), 1–38.
  • [17] K. Hiroe and T. Oshima, “A classification of roots of symmetric Kac–Moody root systems and its application” in Symmetries, Integrable Systems and Representations, Springer Proc. Math. Stat. 40, Springer, Heidelberg, 2013, 195–241.
  • [18] K. Hiroe and D. Yamakawa, Moduli spaces of meromorphic connections and quiver varieties, Adv. Math. 266 (2014), 120–151.
  • [19] M. Inaba, Moduli space of irregular singular parabolic connections of generic ramified type on a smooth projective curve, preprint, arXiv:1606.02369v2 [math.AG].
  • [20] M. Inaba and M.-H. Saito, Moduli of unramified irregular singular parabolic connections on a smooth projective curve, Kyoto J. Math. 53 (2013), 433–482.
  • [21] M. Jimbo, T. Miwa, and K. Ueno, Monodromy preserving deformations of linear ordinary differential equations with rational coefficients, I: General theory and $\tau$-function, Phys. D 2 (1981), 306–352.
  • [22] V. G. Kac, “Root systems, representations of quivers and invariant theory” in Invariant Theory (Montecatini, 1982), Lecture Notes in Math. 996, Springer, Berlin, 1983, 74–108.
  • [23] V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990.
  • [24] N. M. Katz, Rigid Local Systems, Ann. of Math. Stud. 139, Princeton Univ. Press, Princeton, 1996.
  • [25] H. Kawakami, Generalized Okubo systems and the middle convolution, Int. Math. Res. Not. IMRN 2010, no. 17, 3394–3421.
  • [26] A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515–530.
  • [27] V. P. Kostov, The Deligne-Simpson problem—a survey, J. Algebra 281 (2004), 83–108.
  • [28] V. P. Kostov, Additive Deligne-Simpson problem for non-Fuchsian systems, Funkcial. Ekvac. 53 (2010), 395–410.
  • [29] H. Kraft and C. Riedtmann, “Geometry of representations of quivers” in Representations of Algebras (Durham, England, 1985), London Math. Soc. Lecture Note Ser. 116, Cambridge Univ. Press, Cambridge, 1986, 109–145.
  • [30] D. Mumford, J. Forgaty, and F. Kirwan, Geometric Invariant Theory, 3rd ed. Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin, 1994.
  • [31] H. Nakajima, Instanton on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), 365–416.
  • [32] H. Nakajima, Reflection functors for quiver varieties and Weyl group actions, Math. Ann. 327 (2003), 671–721.
  • [33] C. T. Simpson, “Products of matrices” in Differential Geometry, Global Analysis, and Topology (Halifax, 1990), CMS Conf. Proc. 12, Amer. Math. Soc. Providence, 1991, 157–185.
  • [34] K. Takemura, “Introduction to middle convolution for differential equations with irregular singularities” in New Trends in Quantum Integrable Systems, World Sci., Hackensack, N.J., 2011, 393–420.
  • [35] H. Völklein, The braid group and linear rigidity, Geom. Dedicata 84 (2001), 135–150.
  • [36] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Pure Appl. Math. 14, Interscience, New York, 1965.
  • [37] D. Yamakawa, Geometry of multiplicative preprojective algebra, Int. Math. Res. Pap. IMRP 2008, no. rpn008.
  • [38] D. Yamakawa, Quiver varieties with multiplicities, Weyl groups of non-symmetric Kac-Moody algebras, and Painlevé equations, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), no. 087.
  • [39] D. Yamakawa, Middle convolution and Harnad duality, Math. Ann. 349 (2011), 215–262.