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) no. 2, pp. 699-737.

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 ${\mathrm{PSL}}_2\left(\mathbb{Z}\left[\sqrt{3}\right]\right)$ sur le bidisque.

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 ${\mathrm{PSL}}_2\left(\mathbb{Z}\left[\sqrt{3}\right]\right)$ on the bidisc.

[1] Andreev, F. V.; Kitaev, A. V. Transformations $R{S}_4^2\left(3\right)$ of the ranks $\le 4$ and algebraic solutions of the sixth Painlevé equation, Comm. Math. Phys., Volume 228 (2002) no. 1, pp. 151-176 | DOI | MR | Zbl

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

[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 | Zbl

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

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

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

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

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

[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., Volume 7 (2007) no. 1, p. 21-54, 166 | MR | Zbl

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

[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 | DOI | MR | Zbl

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

[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., Volume 18 (2007) no. 6, pp. 723-747 | DOI | MR | Zbl

[25] McQuillan, Michael Canonical models of foliations, Pure Appl. Math. Q., Volume 4 (2008) no. 3, part 2, pp. 877-1012 | DOI | MR | Zbl

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

[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., Volume 209 (1974), pp. 211-248 | DOI | MR | Zbl

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

[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), Volume 77 (1963), pp. 33-71 | DOI | MR | Zbl

[32] Touzet, Frédéric Sur les feuilletages holomorphes transversalement projectifs, Ann. Inst. Fourier (Grenoble), Volume 53 (2003) no. 3, pp. 815-846 | DOI | Numdam | MR | Zbl