Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI

Cousin, Gaël

Annales de l'Institut Fourier, Tome 64 (2014), p. 699-737 / Harvested from Numdam

Résumé

On peut construire facilement des exemples de connexions plates de rang $2$ sur $\mathbb{P}^2$ comme tirés en arrière de connexions sur $\mathbb{P}^1$. On donne un exemple de connexion qui ne peut être obtenue de cette manière. Cet exemple est construit à partir d’une solution algébrique de l’équation de Painlevé VI. On en déduit un feuilletage modulaire. La preuve de ce fait repose sur la classification des feuilletages sur les surfaces projectives par leurs dimensions de Kodaira, fruit du travail de Brunella, McQuillan et Mendes. On décrit ensuite le feuilletage dual. Par une analyse fine de monodromie, on voit que notre surface bifeuilletée est revêtue par la surface modulaire de Hilbert construite en faisant agir $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{3}])$ sur le bidisque.

Abstract

One can easily give examples of rank $2$ flat connections over $\mathbb{P}^2$ by rational pull-back of connections over $\mathbb{P}^1$. We give an example of a connection that can not occur in this way; this example is constructed from an algebraic solution of Painlevé VI equation. From this example we deduce a Hilbert modular foliation. The proof of this relies on the classification of foliations on projective surfaces due to Brunella, Mc Quillan and Mendes. Then, we get the dual foliation and, by a precise monodromy analysis, we see that our twice foliated surface is covered by the classical Hilbert modular surface constructed from the action of $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{3}])$ on the bidisc.

Détail

Publié le : 2014-01-01

MR 3330920 | Zbl 06387290

DOI : https://doi.org/10.5802/aif.2863
Classification:  34M56,  14G35,  37F75,  34M35
Mots clés: feuilletages holomorphes, dimension de Kodaira, surfaces modulaires de Hilbert, connexions plates, équation de Painlevé VI.

Bibliographie

[1] Andreev, F. V.; Kitaev, A. V. Transformations $RS^2_4(3)$ of the ranks $\le 4$ and algebraic solutions of the sixth Painlevé equation, Comm. Math. Phys., Tome 228 (2002) no. 1, pp. 151-176 | Article | MR 1911252 | Zbl 1019.34086

[2] Baily, W. L. Jr.; Borel, A. Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2), Tome 84 (1966), pp. 442-528 | Article | MR 216035 | Zbl 0154.08602

[3] Birman, Joan S. Braids, links, and mapping class groups, Princeton University Press, Princeton, N. J. (1975), pp. 228 (Based on lecture notes by James Cannon) | MR 375281 | Zbl 0305.57013

[4] Boalch, Philip From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. London Math. Soc. (3), Tome 90 (2005) no. 1, pp. 167-208 | Article | MR 2107041 | Zbl 1070.34123

[5] Boalch, Philip The fifty-two icosahedral solutions to Painlevé VI, J. Reine Angew. Math., Tome 596 (2006), pp. 183-214 | Article | MR 2254812 | Zbl 1112.34072

[6] Boalch, Philip Some explicit solutions to the Riemann-Hilbert problem, Differential equations and quantum groups, Eur. Math. Soc., Zürich (IRMA Lect. Math. Theor. Phys.) Tome 9 (2007), pp. 85-112 | MR 2322328

[7] Brunella, Marco Birational geometry of foliations, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Monografí as de Matemática. [Mathematical Monographs] (2000), pp. 138 (disponible en ligne à http://www.impa.br/Publicacoes/Monografias/Abstracts/brunella.ps) | MR 1948251 | Zbl 1073.14022

[8] Brunella, Marco Subharmonic variation of the leafwise Poincaré metric, Invent. Math., Tome 152 (2003) no. 1, pp. 119-148 | Article | MR 1965362 | Zbl 1029.32014

[9] Cerveau, Dominique; Lins-Neto, Alcides; Loray, Frank; Pereira, Jorge Vitório; Touzet, Frédéric Complex codimension one singular foliations and Godbillon-Vey sequences, Mosc. Math. J., Tome 7 (2007) no. 1, p. 21-54, 166 | MR 2324555 | Zbl 1135.37019

[10] Churchill, R. C. Two generator subgroups of ${\rm SL}(2,\mathbb{C})$ and the hypergeometric, Riemann, and Lamé equations, J. Symbolic Comput., Tome 28 (1999) no. 4-5, pp. 521-545 (Differential algebra and differential equations) | Article | MR 1731936 | Zbl 0958.34074

[11] Cogolludo-Agustín, José Ignacio Braid monodromy of algebraic curves, Ann. Math. Blaise Pascal, Tome 18 (2011) no. 1, pp. 141-209 | Article | Numdam | MR 2830090 | Zbl 1254.32043

[12] Corlette, Kevin; Simpson, Carlos On the classification of rank-two representations of quasiprojective fundamental groups, Compos. Math., Tome 144 (2008) no. 5, pp. 1271-1331 | Article | MR 2457528 | Zbl 1155.58006

[13] De La Harpe, P.; Siegfried, P. Singularités de Klein, Enseign. Math. (2), Tome 25 (1979) no. 3-4, p. 207-256 (1980) | MR 570310

[14] Diarra, Karamoko Construction et classification de certaines solutions algébriques des systèmes de Garnier (2012) (Preprint arXiv :1201.1499) | Zbl 1268.34186

[15] Dubrovin, B.; Mazzocco, M. Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math., Tome 141 (2000) no. 1, pp. 55-147 | Article | MR 1767271 | Zbl 0960.34075

[16] Van Der Geer, Gerard Hilbert modular surfaces, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 16 (1988), pp. x+291 | MR 930101 | Zbl 0634.14022

[17] Griffiths, Phillip; Harris, Joseph Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York (1978), pp. xii+813 (Pure and Applied Mathematics) | MR 507725 | Zbl 0836.14001

[18] Hirzebruch, F. Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen, Math. Ann., Tome 126 (1953), pp. 1-22 | Article | MR 62842 | Zbl 0093.27605

[19] Hirzebruch, F. Hilbert modular surfaces, Enseignement Math. (2), Tome 19 (1973), pp. 183-281 | MR 393045 | Zbl 0285.14007

[20] Hirzebruch, F.; Zagier, D. Classification of Hilbert modular surfaces, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo (1977), pp. 43-77 | MR 480356 | Zbl 0354.14011

[21] Inaba, Michi-Aki; Iwasaki, Katsunori; Saito, Masa-Hiko Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence, Int. Math. Res. Not. (2004) no. 1, pp. 1-30 | Article | MR 2036953 | Zbl 1087.34062

[22] Iwasaki, Katsunori; Kimura, Hironobu; Shimomura, Shun; Yoshida, Masaaki From Gauss to Painlevé, Friedr. Vieweg & Sohn, Braunschweig, Aspects of Mathematics, E16 (1991), pp. xii+347 (A modern theory of special functions) | MR 1118604 | Zbl 0743.34014

[23] Lisovyy, Oleg; Tykhyy, Yuriy Algebraic solutions of the sixth Painlevé equation (2008) (Preprint arXiv :0809.4873v2)

[24] Loray, Frank; Pereira, Jorge Vitório Transversely projective foliations on surfaces : existence of minimal form and prescription of monodromy, Internat. J. Math., Tome 18 (2007) no. 6, pp. 723-747 | Article | MR 2337401 | Zbl 1124.37028

[25] Mcquillan, Michael Canonical models of foliations, Pure Appl. Math. Q., Tome 4 (2008) no. 3, part 2, pp. 877-1012 | Article | MR 2435846 | Zbl 1166.14010

[26] Mendes, L. G.; Pereira, J. V. Hilbert modular foliations on the projective plane, Comment. Math. Helv., Tome 80 (2005) no. 2, pp. 243-291 | Article | MR 2142243 | Zbl 1084.32025

[27] Novikov, Dmitry; Yakovenko, Sergei Lectures on meromorphic flat connexions, Disponible en ligne (2002)

[28] Riemenschneider, Oswald Deformationen von Quotientensingularitäten (nach zyklischen Gruppen), Math. Ann., Tome 209 (1974), pp. 211-248 | Article | MR 367276 | Zbl 0275.32010

[29] Scárdua, B. Azevedo Transversely affine and transversely projective holomorphic foliations, Ann. Sci. École Norm. Sup. (4), Tome 30 (1997) no. 2, pp. 169-204 | Article | Numdam | MR 1432053 | Zbl 0889.32031

[30] Shimada, Ichiro Lectures on Zariski-Van Kampen Theorem, Notes de Cours, disponibles en ligne

[31] Shimizu, Hideo On discontinuous groups operating on the product of the upper half planes, Ann. of Math. (2), Tome 77 (1963), pp. 33-71 | Article | MR 145106 | Zbl 0218.10045

[32] Touzet, Frédéric Sur les feuilletages holomorphes transversalement projectifs, Ann. Inst. Fourier (Grenoble), Tome 53 (2003) no. 3, pp. 815-846 http://aif.cedram.org/item?id=AIF_2003__53_3_815_0 | Article | Numdam | MR 2008442 | Zbl 1032.32020