Geometry and braiding of Stokes data; Fission and wild character varieties

Pages 301-365 from Volume 179 (2014), Issue 1 by P. P. Boalch

Abstract

A family of new algebraic Poisson varieties will be constructed, generalising the complex character varieties of Riemann surfaces. Then the well-known (Poisson) mapping class group actions on the character varieties will be generalised.

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      }
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      AUTHOR = {Deligne, Pierre},
      TITLE = {\'{E}quations Différentielles à Points Singuliers Réguliers},
      SERIES = {Lecture Notes in Math},
      VOLUME={163},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1970},
      PAGES = {iii+133},
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      MRNUMBER = {0417174},
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      AUTHOR = {Deligne, Pierre and Malgrange, Bernard and Ramis, Jean-Pierre},
      TITLE = {Singularités Irrégulières},
      SERIES = {Doc. Math. (Paris)},
      VOLUME={5},
      PUBLISHER = {Soc. Math. France},
      ADDRESS = {Paris},
      YEAR = {2007},
      PAGES = {xii+188},
      ISBN = {978-2-85629-241-9},
      MRCLASS = {32S40 (01A60 34-06 34M35 34M40)},
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      }
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      AUTHOR = {Dr{é}zet, Jean-Marc},
      TITLE = {Luna's slice theorem and applications},
      BOOKTITLE = {Algebraic Group Actions and Quotients},
      PAGES = {39--89},
      PUBLISHER = {Hindawi Publ. Corp., Cairo},
      YEAR = {2004},
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      }
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      AUTHOR = {Dubrovin, Boris},
      TITLE = {Geometry of {$2$}{D} topological field theories},
      BOOKTITLE = {Integrable Systems and Quantum Groups},
      VENUE={{M}ontecatini {T}erme, 1993},
      SERIES = {Lecture Notes in Math.},
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      MRCLASS = {58D29 (14N10 20F55 32G34 58F07 81T40)},
      MRNUMBER = {1397274},
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      }
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      AUTHOR = {Flaschka, H. and Newell, A. C.},
      TITLE = {The inverse monodromy transform is a canonical transformation},
      BOOKTITLE = {Nonlinear Problems: Present and Future},
      VENUE={{L}os {A}lamos, {N}.{M}., 1981},
      SERIES = {North-Holland Math. Stud.},
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      PAGES = {65--89},
      PUBLISHER = {North-Holland},
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      YEAR = {1982},
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      MRREVIEWER = {D. ter Haar},
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     }
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      AUTHOR = {Gelfand, Sergei and MacPherson, Robert and Vilonen, Kari},
      TITLE = {Perverse sheaves and quivers},
      JOURNAL = {Duke Math. J.},
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      MRCLASS = {14F32 (32S60)},
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      }
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      PUBLISHER = {Princeton Univ. Press},
      SERIES = {Math. Notes},
      ADDRESS = {Princeton, NJ},
      YEAR = {1967},
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      MRCLASS = {57.30 (30.00)},
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      AUTHOR = {Hitchin, N. J.},
      TITLE = {The self-duality equations on a {R}iemann surface},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {55},
      YEAR = {1987},
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      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {32G13 (14F05 14H15 32L10 53C05 58E99 81E13)},
      MRNUMBER = {0887284},
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      }
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      author = {Hitchin, N. J.},
      TITLE = {Frobenius manifolds},
      BOOKTITLE={Gauge Theory and Symplectic Geometry},
      EDITOR={Hurtubise, J. and Lalonde, F.},
      SERIES={NATO ASI Series C: Math \& Phys.},
      VOLUME={488},
      PUBLISHER = {Kluwer Acad., Publ.},
      ADDRESS = {Dordrecht},
      YEAR = {1997},
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      DOI = {10.1007/978-94-017-1667-3_3},
     }
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      AUTHOR = {Jimbo, Michio and Miwa, Tetsuji and Ueno, Kimio},
      TITLE = {Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. {I}. {G}eneral theory and {$\tau $}-function},
      JOURNAL = {Phys. D},
      FJOURNAL = {Physica D. Nonlinear Phenomena},
      VOLUME = {2},
      YEAR = {1981},
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      ISSN = {0167-2789},
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      MRNUMBER = {0630674},
      MRREVIEWER = {V. A. Golubeva},
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     }
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      AUTHOR = {Johnson, Dennis and Millson, John J.},
      TITLE = {Deformation spaces associated to compact hyperbolic manifolds},
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      VENUE={{N}ew {H}aven, {C}onn., 1984},
      SERIES = {Progr. Math.},
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      PUBLISHER = {Birkhäuser},
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      MRCLASS = {34A20 (34-02)},
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      AUTHOR = {Kazhdan, D. and Lusztig, G.},
      TITLE = {Fixed point varieties on affine flag manifolds},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {62},
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      NUMBER = {2},
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      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {14L30 (20G25 32M10)},
      MRNUMBER = {0947819},
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      AUTHOR = {King, A. D.},
      TITLE = {Moduli of representations of finite-dimensional algebras},
      JOURNAL = {Quart. J. Math. Oxford Ser.},
      FJOURNAL = {The Quarterly Journal of Mathematics. Oxford. Second Series},
      VOLUME = {45},
      YEAR = {1994},
      NUMBER = {180},
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      ISSN = {0033-5606},
      CODEN = {QJMAAT},
      MRCLASS = {16G10 (14D25)},
      MRNUMBER = {1315461},
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      AUTHOR = {Kostov, Vladimir P.},
      TITLE = {Additive {D}eligne-{S}impson problem for non-{F}uchsian systems},
      JOURNAL = {Funkcial. Ekvac.},
      FJOURNAL = {Funkcialaj Ekvacioj. Serio Internacia},
      VOLUME = {53},
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      ISSN = {0532-8721},
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      AUTHOR = {Krichever, I.},
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      JOURNAL = {Mosc. Math. J.},
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      MRNUMBER = {1986088},
      MRREVIEWER = {Henrik Aratyn},
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      AUTHOR = {Loday-Richaud, Mich{è}le},
      TITLE = {Stokes phenomenon, multisummability and differential {G}alois groups},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
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      }
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      AUTHOR={Luna, D.},
      TITLE={Slices étales},
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      SERIES = {Bull. Soc. Math. France M{é}m.},
      VOLUME={33},
      PUBLISHER = {Société Mathématique de France},
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    @incollection {maisonobe-couple, MRKEY = {1018867},
      AUTHOR = {Maisonobe, P.},
      TITLE = {Faisceaux pervers sur {${\bf C}$} relativement à {$\{0\}$} et couple {$u:E{\rightleftarrows}F:v$}},
      BOOKTITLE = {Introduction à la Théorie Algébrique des Systèmes Différentiels},
      VENUE={{P}lans-sur-{B}ex, 1984},
      SERIES = {Travaux en Cours},
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    @incollection {Mal79, MRKEY = {0548145},
      AUTHOR = {Malgrange, B.},
      TITLE = {Remarques sur les équations différentielles à points singuliers irréguliers},
      BOOKTITLE = {\'{E}quations différentielles et systèmes de {P}faff dans le champ complexe ({S}em., {I}nst. {R}ech. {M}ath. {A}vancée, {S}trasbourg, 1975)},
      SERIES = {Lecture Notes in Math.},
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      MRNUMBER = {0548145},
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     }
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      VOLUME = {96},
      PUBLISHER = {Birkhäuser},
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      MRCLASS = {32C38 (34A20 35A27)},
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      AUTHOR = {Nakajima, Hiraku},
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Keywords
, , , ,
DOI
https://doi.org/10.4007/annals.2014.179.1.5
MR
3126570
Zbl
06260589
Milestones
Received: 15 December 2011
Revised: 7 September 2012
Accepted: 6 November 2012
Published online: 1 January 2014

Authors

P. P. Boalch

École Normale Supérieure and CNRS, Paris, France