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Séminaires et Congrès - Titles - 27 (2013) 115-168

Titles < 27

Geometric and differential Galois theories
D. Bertrand, Ph. Boalch, J-M. Couveignes, P. Dèbes
Séminaires et Congrès 27 (2013), xviii + 247 pages
Presentation, Summary

Foliations on the moduli space of rank two connections on the projective line minus four points
Frank Loray, Masa-Hiko Saito, Carlos Simpson
Séminaires et Congrès 27 (2013), 115-168

Résumé :
Feuilletages de l'espace des modules des connexions de rang 2 sur la sphère privée quatre points
Nous considérons certains feuilletages naturels de l'espace des modules (de Painlevé VI) des connexions de rang 2 sur ^1 - t_1,t_2,t_3,t_4. Ces feuilletages sont des fibrations et s'interprètent en termes de filtration de Hodge non abélienne, donnant une preuve de la conjecture de feuilletage dans ce cadre. Essentiellement deux sortes de fibrations apparaissent : elles proviennent respectivement des singularités apparentes et de la structure sous-jacente de fibré quasiparabolique. Nous montrons qu'elles sont transverses. La symétrie d'Okamoto, qui peut-être vue comme la convolution moyenne de Katz, échange les deux types de feuilletages (singularité apparente et fibré quasiparabolique).

Abstract:
We look at natural foliations on the Painlevé VI moduli space of regular connections of rank 2 on ^1 - t_1,t_2,t_3,t_4. These foliations are fibrations, and are interpreted in terms of the nonabelian Hodge filtration, giving a proof of the nonabelian Hodge foliation conjecture in this case. Two basic kinds of fibrations arise: from apparent singularities, and from quasiparabolic bundles. We show that these are transverse. Okamoto's additional symmetry, which may be seen as Katz's middle convolution, exchanges the quasiparabolic and apparent-singularity foliations.

Keywords: Painlevé VI, Representation, Fundamental group, Logarithmic connection, Higgs bundle, Parabolic structure, Projective line, apparent singularity, Okamoto-Painlevé pair, Middle convolution

Class. math. : Primary 34M55; Secondary 14D20, 32G20, 32G34


ISSN : 1285-2783

Bibliographie:

1
Aidan, J.
Doctoral dissertation, in preparation
Jussieu (2007)
2
Arinkin, D.
Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on P^1 minus 4 points
Selecta Math. (N.S.) 7 (2001) 213–239
Math Reviews MR1860014 (2003a:14010)
3
Arinkin, D. and Lysenko, S.
On the moduli of SL(2)-bundles with connections on P^1x_1,,x_4
Int. Math. Res. Not. 1997 (1997) 983–999
Math Reviews MR1488348 (99c:14013)
4
Arinkin, D. and Lysenko, S.
Isomorphisms between moduli spaces of SL(2)-bundles with connections on P^1 x_1,, x_4
Math. Res. Lett. 4 (1997) 181–190
Math Reviews MR1453052 (98h:14012)
5
Atiyah, M. F.
Complex analytic connections in fibre bundles
Trans. Amer. Math. Soc. 85 (1957) 181–207
Math Reviews MR0086359 (19,172c)
6
Biswas, Indranil
A criterion for the existence of a flat connection on a parabolic vector bundle
Adv. Geom. 2 (2002) 231–241
Math Reviews MR1924757 (2003f:14034)
7
Boalch, Philip
Painlevé equations and complex reflections
in Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002)
Ann. Inst. Fourier (Grenoble) 53 (4) (2003) 1009–1022
Math Reviews MR2033508 (2005a:34128)
8
Boalch, Philip
From Klein to Painlevé via Fourier, Laplace and Jimbo
Proc. London Math. Soc. 90 (2005) 167–208
Math Reviews MR2107041 (2005h:34233)
9
Boalch, Philip
Six results on Painlevé VI
in Théories asymptotiques et équations de Painlevé
Sémin. Congr. 14 (2006) 1–20
Math Reviews MR2349685 (2008i:34162)
10
Boden, Hans U. and Yokogawa, Kôji
Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves. I
Internat. J. Math. 7 (1996) 573–598
Math Reviews MR1411302 (98a:14017)
11
Borne, Niels
Sur les représentations du groupe fondamental d'une variété privée d'un diviseur à croisements normaux simples
Indiana Univ. Math. J. 58 (2009) 137–180
Math Reviews MR2504408 (2010h:14069)
12
13
Cantat, Serge and Loray, Frank
Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation
Ann. Inst. Fourier (Grenoble) 59 (2009) 2927–2978
Math Reviews MR2649343 (2011c:37105)
14
Crawley-Boevey, William
Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity
Publ. Math. Inst. Hautes Études Sci. 100 (2004) 171–207
Math Reviews MR2102700 (2005g:14067)
15
Deligne, Pierre
Équations différentielles à points singuliers réguliers
Lecture Notes in Math., vol. 163, Springer, 1970
Math Reviews MR0417174 (54 \#5232)
16
Dettweiler, Michael and Reiter, Stefan
Painlevé equations and the middle convolution
Adv. Geom. 7 (2007) 317–330
Math Reviews MR2334802 (2008d:34165)
17
Dubrovin, B. and Mazzocco, Marta
Monodromy of certain Painlevé-VI transcendents and reflection groups
Invent. Math. 141 (2000) 55–147
Math Reviews MR1767271 (2001j:34114)
18
Esnault, Hélène and Viehweg, Eckart
Semistable bundles on curves and irreducible representations of the fundamental group
in Algebraic geometry: Hirzebruch 70 (Warsaw, 1998)
Contemp. Math. 241 (1999) 129–138
Math Reviews MR1718141 (2000i:14049)
19
García-Prada, O. and Gothen, P. B. and Muñoz, V.
Betti numbers of the moduli space of rank 3 parabolic Higgs bundles
Mem. Amer. Math. Soc. 187 (2007) 80
Math Reviews MR2308696 (2008a:14020)
20
21
Hitchin, N. J.
The self-duality equations on a Riemann surface
Proc. London Math. Soc. 55 (1987) 59–126
Math Reviews MR887284 (89a:32021)
22
Inaba, Michiaki and Iwasaki, Katsunori and Saito, Masa-Hiko
Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I
Publ. Res. Inst. Math. Sci. 42 (2006) 987–1089
Math Reviews MR2289083 (2008k:32040)
23
Inaba, Michiaki and Iwasaki, Katsunori and Saito, Masa-Hiko
Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II
in Moduli spaces and arithmetic geometry
Adv. Stud. Pure Math. 45 (2006) 387–432
Math Reviews MR2310256 (2009j:14015)
24
Inaba, Michiaki and Iwasaki, Katsunori and Saito, Masa-Hiko
Dynamics of the sixth Painlevé equation
in Théories asymptotiques et équations de Painlevé
Sémin. Congr. 14 (2006) 103–167
Math Reviews MR2353464 (2008j:14022)
25
Iyer, Jaya N. and Simpson, Carlos T.
The Chern character of a parabolic bundle, and a parabolic corollary of Reznikov's theorem
in Geometry and dynamics of groups and spaces
Progr. Math. 265 (2008) 439–485
Math Reviews MR2402414 (2010d:14005)
26
Jimbo, Michio and Miwa, Tetsuji
Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II
Phys. D 2 (1981) 407–448
Math Reviews MR625446 (83k:34010b)
27
Konno, Hiroshi
Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface
J. Math. Soc. Japan 45 (1993) 253–276
Math Reviews MR1206652 (94g:32027)
28
Loray, Frank
Okamoto symmetry of Painlevé VI equation and isomonodromic deformation of Lamé connections
in Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies
RIMS Kôkyûroku Bessatsu, B2 (2007) 129–136
Math Reviews MR2310025 (2008c:34189)
29
Machu, Francois-Xavier
Monodromy of a class of logarithmic connections on an elliptic curve
SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007) Paper 082, 31
Math Reviews MR2366940 (2009a:14018)
30
Maruyama, M. and Yokogawa, Kôji
Moduli of parabolic stable sheaves
Math. Ann. 293 (1992) 77–99
Math Reviews MR1162674 (93d:14022)
31
Mazzocco, Marta
Irregular isomonodromic deformations for Garnier systems and Okamoto's canonical transformations
J. London Math. Soc. 70 (2004) 405–419
Math Reviews MR2078901 (2005e:32027)
32
Mochizuki, Takuro
Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D-modules. I and II
Mem. Amer. Math. Soc. 185 (2007) 324
Math Reviews MR2281877 (2007j:32028a)
33
Mochizuki, Takuro
Kobayashi-Hitchin correspondence for tame harmonic bundles and an application
Astérisque 309 (2006) 117
Math Reviews MR2310103 (2008f:32029)
34
Nakajima, Hiraku
Hyper-Kähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces
in Moduli of vector bundles (Sanda, 1994; Kyoto, 1994)
Lecture Notes in Pure and Appl. Math. 179 (1996) 199–208
Math Reviews MR1397988 (97k:32030)
35
Nitsure, Nitin
Moduli of semistable logarithmic connections
J. Amer. Math. Soc. 6 (1993) 597–609
Math Reviews MR1182671 (93i:32025)
36
Noumi, Masatoshi and Yamada, Yasuhiko
A new Lax pair for the sixth Painlevé equation associated with so(8)
in Microlocal analysis and complex Fourier analysis
(2002) 238–252
Math Reviews MR2068540 (2005b:34191)
37
Oblezin, Sergey
Isomonodromic deformations of sl(2) Fuchsian systems on the Riemann sphere
Mosc. Math. J. 5 (2005) 415–441, 494–495
Math Reviews MR2200759 (2007g:32012)
38
Okamoto, Kazuo
Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé
Japan. J. Math. (N.S.) 5 (1979) 1–79
Math Reviews MR614694 (83m:58005)
39
Okamoto, Kazuo
Isomonodromic deformation and Painlevé equations, and the Garnier system
J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 (1986) 575–618
Math Reviews MR866050 (88e:58042)
40
Okamoto, Kazuo
Studies on the Painlevé equations. I. Sixth Painlevé equation P_VI
Ann. Mat. Pura Appl. 146 (1987) 337–381
Math Reviews MR916698 (88m:58062)
41
Sabbah, Claude
Polarizable twistor D-modules
Astérisque 300 (2005) 208
Math Reviews MR2156523 (2006d:32009)
42
Saito, Masa-Hiko and Takebe, Taro and Terajima, Hitomi
Deformation of Okamoto-Painlevé pairs and Painlevé equations
J. Algebraic Geom. 11 (2002) 311–362
Math Reviews MR1874117 (2002j:34158)
43
Sakai, Hidetaka
Rational surfaces associated with affine root systems and geometry of the Painlevé equations
Comm. Math. Phys. 220 (2001) 165–229
Math Reviews MR1882403 (2003c:14030)
44
Schmitt, Alexander
Projective moduli for Hitchin pairs
Internat. J. Math. 9 (1998) 107–118
Math Reviews MR1612251 (99e:14011)
45
Seshadri, C. S.
Moduli of -vector bundles over an algebraic curve
in Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969)
(1970) 139–260
Math Reviews MR0280496 (43 \#6216)
46
Simpson, Carlos T.
Harmonic bundles on noncompact curves
J. Amer. Math. Soc. 3 (1990) 713–770
Math Reviews MR1040197 (91h:58029)
47
Simpson, Carlos T.
Higgs bundles and local systems
Publ. Math. I.H.É.S. 75 (1992) 5–95
Math Reviews MR1179076 (94d:32027)
48
Simpson, Carlos
Katz's middle convolution algorithm
Pure Appl. Math. Q. 5 (2009) 781–852
Math Reviews MR2508903 (2010a:14033)
49
Simpson, Carlos
Iterated destabilizing modifications for vector bundles with connection
in Vector bundles and complex geometry
Contemp. Math. 522 (2010) 183–206
Math Reviews MR2681730 (2011h:14040)
50
Szabó, Szilárd
Deformations of Fuchsian equations and logarithmic connections
Arxiv preprint math/0703230 (2007)
51
Weil, André
Généralisation des fonctions abéliennes
J. Math. Pures Appl. 17 (1938) 47–87
52
Yamakawa, Daisuke
Middle convolution and Harnad duality
Math. Ann. 349 (2011) 215–262
Math Reviews MR2753801
53
Yokogawa, Kôji
Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves
J. Math. Kyoto Univ. 33 (1993) 451–504
Math Reviews MR1231753 (94h:14013)