Geometry and braiding of Stokes data; fission and wild character varieties. (English) Zbl 1283.53075

Boalch, P. P.

Ann. Math. (2) 179, No. 1, 301-365 (2014).

This paper constructs new algebraic Poisson varieties which generalize the complex varieties of Riemann surfaces. The author starts with the background definitions and properties. The spaces of Stokes data are introduced. The author proves that the space of Stokes data attached to a global irregular curve is an algebraic Poisson variety. Finally, it is shown that the family of Poisson varieties corresponding to admissible irregular curves fit together into a Poisson local system. The usual Poisson mapping class group actions on the character varieties are also generalized.

Reviewer: Angela Gammella-Mathieu (Metz)

MSC:

53D17	Poisson manifolds; Poisson groupoids and algebroids
53D18	Generalized geometries (à la Hitchin)

Keywords:

character varieties; irregular curves; Stokes data; symplectic structures; algebraic Poisson varieties

References:

[1]	A. Alekseev, H. Bursztyn, and E. Meinrenken, ”Pure spinors on Lie groups,” in From Probability to Geometry I. Volume in honor of the 60th birthday of Jean-Michel Bismut, Dai, X., Léandre, R., Ma, X., and Zhang, W., Eds., , 2009, vol. 327, pp. 131-199. · Zbl 1251.53052
[2]	A. Alekseev, Y. Kosmann-Schwarzbach, and E. Meinrenken, ”Quasi-Poisson manifolds,” Canad. J. Math., vol. 54, iss. 1, pp. 3-29, 2002. · Zbl 1006.53072
[3]	A. Alekseev, A. Malkin, and E. Meinrenken, ”Lie group valued moment maps,” J. Differential Geom., vol. 48, iss. 3, pp. 445-495, 1998. · Zbl 0948.53045
[4]	M. F. Atiyah and R. Bott, ”The Yang-Mills equations over Riemann surfaces,” Philos. Trans. Roy. Soc. London Ser. A, vol. 308, iss. 1505, pp. 523-615, 1983. · Zbl 0509.14014
[5]	M. Audin, ”Lectures on gauge theory and integrable systems,” in Gauge Theory and Symplectic Geometry, Hurtubise, J. and Lalonde, F., Eds., Dordrecht: Kluwer Acad. Publ., 1997, vol. 488, pp. 1-48. · Zbl 0873.58016
[6]	D. G. Babbitt and V. S. Varadarajan, Local Moduli for Meromorphic Differential Equations, Paris: Math. Soc. France, 1989, vol. 169-170. · Zbl 0683.34003
[7]	W. Balser, B. L. J. Braaksma, J. -P. Ramis, and Y. Sibuya, ”Multisummability of formal power series solutions of linear ordinary differential equations,” Asymptotic Anal., vol. 5, iss. 1, pp. 27-45, 1991. · Zbl 0754.34057
[8]	W. Balser, W. B. Jurkat, and D. A. Lutz, ”Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations,” J. Math. Anal. Appl., vol. 71, iss. 1, pp. 48-94, 1979. · Zbl 0415.34008
[9]	W. Balser, W. B. Jurkat, and D. A. Lutz, ”A general theory of invariants for meromorphic differential equations. II. Proper invariants,” Funkcial. Ekvac., vol. 22, iss. 3, pp. 257-283, 1979. · Zbl 0473.34028
[10]	O. Biquard and P. P. Boalch, ”Wild non-abelian Hodge theory on curves,” Compos. Math., vol. 140, iss. 1, pp. 179-204, 2004. · Zbl 1051.53019
[11]	P. P. Boalch, Global Weyl groups and a new theory of multiplicative quiver varieties, 2013. · Zbl 1332.14059
[12]	P. P. Boalch, ”Stokes matrices, Poisson Lie groups and Frobenius manifolds,” Invent. Math., vol. 146, iss. 3, pp. 479-506, 2001. · Zbl 1044.53060
[13]	P. P. Boalch, ”Symplectic manifolds and isomonodromic deformations,” Adv. Math., vol. 163, iss. 2, pp. 137-205, 2001. · Zbl 1001.53059
[14]	P. P. Boalch, ”\(G\)-bundles, isomonodromy, and quantum Weyl groups,” Int. Math. Res. Not., vol. 2002, p. no. 22, 1129-1166. · Zbl 1003.58028
[15]	P. P. Boalch, ”From Klein to Painlevé via Fourier, Laplace and Jimbo,” Proc. London Math. Soc., vol. 90, iss. 1, pp. 167-208, 2005. · Zbl 1070.34123
[16]	P. P. Boalch, ”Quasi-Hamiltonian geometry of meromorphic connections,” Duke Math. J., vol. 139, iss. 2, pp. 369-405, 2007. · Zbl 1126.53055
[17]	P. P. Boalch, Some geometry of irregular connections on curves, 2007. · Zbl 1126.53055
[18]	P. P. Boalch, Irregular connections and Kac-Moody root systems, 2008.
[19]	P. P. Boalch, ”Through the analytic halo: Fission via irregular singularities,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 59, iss. 7, pp. 2669-2684, 2009. · Zbl 1206.53086
[20]	P. P. Boalch, ”Riemann-Hilbert for tame complex parahoric connections,” Transform. Groups, vol. 16, iss. 1, pp. 27-50, 2011. · Zbl 1232.34117
[21]	P. P. Boalch, ”Simply-laced isomonodromy systems,” Publ. Math. I.H.E.S., vol. 116, pp. 1-68, 2012. · Zbl 1270.34204
[22]	A. I. Bondal, ”A symplectic groupoid of triangular bilinear forms and the braid group,” Izv. Ross. Akad. Nauk Ser. Mat., vol. 68, iss. 4, pp. 19-74, 2004. · Zbl 1084.58502
[23]	A. Borel, Linear Algebraic Groups, Second ed., New York: Springer-Verlag, 1991, vol. 126. · Zbl 0726.20030
[24]	L. Boutet de Monvel, ”\(\mathcalD\)-modules holonômes réguliers en une variable,” in Mathematics and Physics, Boston, MA: Birkhäuser, 1983, vol. 37, pp. 313-321. · Zbl 0578.35080
[25]	H. Cassens and P. Slodowy, ”On Kleinian singularities and quivers,” in Singularities, Basel: Birkhäuser, 1998, vol. 162, pp. 263-288. · Zbl 0957.14004
[26]	S. Cecotti and C. Vafa, ”On classification of \(N=2\) supersymmetric theories,” Comm. Math. Phys., vol. 158, iss. 3, pp. 569-644, 1993. · Zbl 0787.58049
[27]	W. Crawley-Boevey and P. Shaw, ”Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem,” Adv. Math., vol. 201, iss. 1, pp. 180-208, 2006. · Zbl 1095.15014
[28]	C. De Concini, V. G. Kac, and C. Procesi, ”Quantum coadjoint action,” J. Amer. Math. Soc., vol. 5, iss. 1, pp. 151-189, 1992. · Zbl 0747.17018
[29]	P. Deligne, Équations Différentielles à Points Singuliers Réguliers, New York: Springer-Verlag, 1970, vol. 163. · Zbl 0244.14004
[30]	P. Deligne, B. Malgrange, and J. Ramis, Singularités Irrégulières, Paris: Soc. Math. France, 2007, vol. 5. · Zbl 1130.14001
[31]	J. Drézet, ”Luna’s slice theorem and applications,” in Algebraic Group Actions and Quotients, Hindawi Publ. Corp., Cairo, 2004, pp. 39-89. · Zbl 1109.14307
[32]	B. Dubrovin, ”Geometry of \(2\)D topological field theories,” in Integrable Systems and Quantum Groups, New York: Springer-Verlag, 1996, vol. 1620, pp. 120-348. · Zbl 0841.58065
[33]	H. Flaschka and A. C. Newell, ”The inverse monodromy transform is a canonical transformation,” in Nonlinear Problems: Present and Future, Amsterdam: North-Holland, 1982, vol. 61, pp. 65-89. · Zbl 0555.35107
[34]	S. Gelfand, R. MacPherson, and K. Vilonen, ”Perverse sheaves and quivers,” Duke Math. J., vol. 83, iss. 3, pp. 621-643, 1996. · Zbl 0861.32022
[35]	W. M. Goldman, ”The symplectic nature of fundamental groups of surfaces,” Adv. in Math., vol. 54, iss. 2, pp. 200-225, 1984. · Zbl 0574.32032
[36]	R. C. Gunning, Lectures on Vector Bundles over Riemann Surfaces, Princeton, NJ: Princeton Univ. Press, 1967. · Zbl 0163.31903
[37]	N. J. Hitchin, ”The self-duality equations on a Riemann surface,” Proc. London Math. Soc., vol. 55, iss. 1, pp. 59-126, 1987. · Zbl 0634.53045
[38]	N. J. Hitchin, ”Frobenius manifolds,” in Gauge Theory and Symplectic Geometry, Hurtubise, J. and Lalonde, F., Eds., Dordrecht: Kluwer Acad., Publ., 1997, vol. 488. · Zbl 0867.53027
[39]	M. Jimbo, T. Miwa, and K. Ueno, ”Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and \(\tau \)-function,” Phys. D, vol. 2, iss. 2, pp. 306-352, 1981. · Zbl 1194.34167
[40]	D. Johnson and J. J. Millson, ”Deformation spaces associated to compact hyperbolic manifolds,” in Discrete Groups in Geometry and Analysis, Boston, MA: Birkhäuser, 1987, vol. 67, pp. 48-106. · Zbl 0664.53023
[41]	W. B. Jurkat, Meromorphe Differentialgleichungen, New York: Springer-Verlag, 1978, vol. 637. · Zbl 0408.34004
[42]	D. Kazhdan and G. Lusztig, ”Fixed point varieties on affine flag manifolds,” Israel J. Math., vol. 62, iss. 2, pp. 129-168, 1988. · Zbl 0658.22005
[43]	A. D. King, ”Moduli of representations of finite-dimensional algebras,” Quart. J. Math. Oxford Ser., vol. 45, iss. 180, pp. 515-530, 1994. · Zbl 0837.16005
[44]	V. P. Kostov, ”Additive Deligne-Simpson problem for non-Fuchsian systems,” Funkcial. Ekvac., vol. 53, iss. 3, pp. 395-410, 2010. · Zbl 1246.15018
[45]	I. Krichever, ”Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations,” Mosc. Math. J., vol. 2, iss. 4, pp. 717-752, 806, 2002. · Zbl 1044.70010
[46]	M. Loday-Richaud, ”Stokes phenomenon, multisummability and differential Galois groups,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 44, iss. 3, pp. 849-906, 1994. · Zbl 0812.34004
[47]	D. Luna, ”Slices étales,” in Sur les Groupes Algébriques, Paris: Société Mathématique de France, 1973, vol. 33, p. 105. · Zbl 0286.14014
[48]	P. Maisonobe, ”Faisceaux pervers sur \({\mathbf C}\) relativement à \(\{0\}\) et couple \(u:E{\rightleftarrows}F:v\),” in Introduction à la Théorie Algébrique des Systèmes Différentiels, Paris: Hermann, 1988, vol. 34, pp. 135-146. · Zbl 0684.32009
[49]	B. Malgrange, ”Remarques sur les équations différentielles à points singuliers irréguliers,” in Équations différentielles et systèmes de Pfaff dans le champ complexe (Sem., Inst. Rech. Math. Avancée, Strasbourg, 1975), New York: Springer-Verlag, 1979, pp. 77-86. · Zbl 0423.32014
[50]	B. Malgrange, Équations Différentielles à Coefficients Polynomiaux, Boston, MA: Birkhäuser, 1991, vol. 96. · Zbl 0764.32001
[51]	J. Martinet and J. Ramis, ”Elementary acceleration and multisummability. I,” Ann. Inst. H. Poincaré Phys. Théor., vol. 54, iss. 4, pp. 331-401, 1991. · Zbl 0748.12005
[52]	H. Nakajima, ”Hyper-Kähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces,” in Moduli of Vector Bundles, New York: Dekker, 1996, pp. 199-208. · Zbl 0881.14006
[53]	J. Ramis, ”Phénomène de Stokes et resommation,” C. R. Acad. Sci. Paris Sér. I Math., vol. 301, iss. 4, pp. 99-102, 1985. · Zbl 0582.34006
[54]	R. W. Richardson, ”Conjugacy classes of \(n\)-tuples in Lie algebras and algebraic groups,” Duke Math. J., vol. 57, iss. 1, pp. 1-35, 1988. · Zbl 0685.20035
[55]	R. J. Schilling, ”Baker functions for compact Riemann surfaces,” Proc. Amer. Math. Soc., vol. 98, iss. 4, pp. 671-675, 1986. · Zbl 0614.58022
[56]	G. Segal and G. Wilson, ”Loop groups and equations of KdV type,” Inst. Hautes Études Sci. Publ. Math., vol. 61, pp. 5-65, 1985. · Zbl 0592.35112
[57]	Y. Sibuya, ”Stokes phenomena,” Bull. Amer. Math. Soc., vol. 83, iss. 5, pp. 1075-1077, 1977. · Zbl 0386.34008
[58]	A. S. Sikora, ”Character varieties,” Trans. Amer. Math. Soc., vol. 364, iss. 10, pp. 5173-5208, 2012. · Zbl 1291.14022
[59]	C. T. Simpson, ”Harmonic bundles on noncompact curves,” J. Amer. Math. Soc., vol. 3, iss. 3, pp. 713-770, 1990. · Zbl 0713.58012
[60]	C. T. Simpson, ”Moduli of representations of the fundamental group of a smooth projective variety. II,” Inst. Hautes Études Sci. Publ. Math., vol. 80, pp. 5-79, 1995. · Zbl 0891.14006
[61]	S. Szabó, ”Nahm transform for integrable connections on the Riemann sphere,” Mém. Soc. Math. Fr., iss. 110, p. ii, 2007. · Zbl 1213.53005
[62]	M. Ugaglia, ”On a Poisson structure on the space of Stokes matrices,” Internat. Math. Res. Notices, iss. 9, pp. 473-493, 1999. · Zbl 0939.34073
[63]	M. Van den Bergh, ”Double Poisson algebras,” Trans. Amer. Math. Soc., vol. 360, iss. 11, pp. 5711-5769, 2008. · Zbl 1157.53046
[64]	M. Van den Bergh, ”Non-commutative quasi-Hamiltonian spaces,” in Poisson Geometry in Mathematics and Physics, Providence, RI: Amer. Math. Soc., 2008, vol. 450, pp. 273-299. · Zbl 1236.58019
[65]	A. Weil, ”Remarks on the cohomology of groups,” Ann. of Math., vol. 80, pp. 149-157, 1964. · Zbl 0192.12802
[66]	E. Witten, ”Gauge theory and wild ramification,” Anal. Appl. \((\)Singap.\()\), vol. 6, iss. 4, pp. 429-501, 2008. · Zbl 1177.81101
[67]	N. M. J. Woodhouse, ”The symplectic and twistor geometry of the general isomonodromic deformation problem,” J. Geom. Phys., vol. 39, iss. 2, pp. 97-128, 2001. · Zbl 1116.32015
[68]	P. Xu, ”Dirac submanifolds and Poisson involutions,” Ann. Sci. École Norm. Sup., vol. 36, iss. 3, pp. 403-430, 2003. · Zbl 1047.53052
[69]	D. Yamakawa, ”Geometry of multiplicative preprojective algebra,” Int. Math. Res. Pap. IMRP, vol. 2008, p. I. · Zbl 1197.16019