Representations of quasi-projective groups, flat connections and transversely projective foliations

Loray, Frank; Pereira, Jorge Vitório; Touzet, Frédéric

Journal de l’École polytechnique - Mathématiques, Tome 3 (2016), pp. 263-308.

L’objet de cet article est d’établir un théorème de structure pour les feuilletages singuliers transversalement projectifs de codimension $1$ sur une variété projective lisse. Pour ce faire, nous étendons d’abord la classification de Corlette et Simpson de représentations de rang $2$ des groupes fondamentaux des variétés quasi-projectives lisses en omettant l’hypothèse de quasi-unipotence à l’infini. Ensuite, nous établissons une classification analogue pour les connexions méromorphes plates de rang $2$. En particulier, nous montrons qu’une connexion méromorphe plate de rang $2$ avec des singularités irrégulières et des matrices de Stokes non triviales se factorise par une connexion sur une courbe.

The main purpose of this paper is to provide a structure theorem for codimension-one singular transversely projective foliations on projective manifolds. To reach our goal, we firstly extend Corlette-Simpson’s classification of rank-two representations of fundamental groups of quasi-projective manifolds by dropping the hypothesis of quasi-unipotency at infinity. Secondly we establish a similar classification for rank-two flat meromorphic connections. In particular, we prove that a rank-two flat meromorphic connection with irregular singularities having non trivial Stokes matrices projectively factors through a connection over a curve.

Classification : 37F75, 34M40, 32S40
Keywords: Foliation, transverse structure, birational geometry, flat connections, irregular singular points, Stokes matrices
Mot clés : Feuilletage, structure transverse, géométrie birationnelle, connexion plate, points singuliers irréguliers, matrices de Stokes

[1] André, Y. Structure des connexions méromorphes formelles de plusieurs variables et semi-continuité de l’irrégularité, Invent. Math., Volume 170 (2007) no. 1, pp. 147-198 | DOI | Zbl

[2] Artal Bartolo, E.; Cogolludo-Agustín, J. I.; Matei, D. Characteristic varieties of quasi-projective manifolds and orbifolds, Geom. Topol., Volume 17 (2013) no. 1, pp. 273-309 | DOI | MR | Zbl

[3] Berthier, M.; Touzet, F. Sur l’intégration des équations différentielles holomorphes réduites en dimension deux, Bol. Soc. Brasil. Mat. (N.S.), Volume 30 (1999) no. 3, pp. 247-286 | DOI | Zbl

[4] Brunella, M. Minimal models of foliated algebraic surfaces, Bull. Soc. math. France, Volume 127 (1999) no. 2, pp. 289-305 | DOI | Numdam | MR | Zbl

[5] Brunella, M. Birational geometry of foliations, Publicações Matemáticas do IMPA, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2004, iv+138 pages | Zbl

[6] Camacho, C.; Azevedo Scárdua, B. Holomorphic foliations with Liouvillian first integrals, Ergodic Theory Dynam. Systems, Volume 21 (2001) no. 3, pp. 717-756 Erratum: Ibid. 23 (2003), no. 3, p. 985–987 | DOI | MR | Zbl

[7] Cano, F.; Cerveau, D.; Déserti, J. Théorie élémentaires des feuilletages holomorphes singuliers, Échelles, Belin, Paris, 2013

[8] Casale, G. Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 3, pp. 735-779 http://aif.cedram.org/item?id=AIF_2006__56_3_735_0 | DOI | Numdam | Zbl

[9] Cerveau, D.; Lins-Neto, A.; Loray, F.; Pereira, J. V.; Touzet, F. Complex codimension one singular foliations and Godbillon-Vey sequences, Moscow Math. J., Volume 7 (2007) no. 1, p. 21-54, 166 | DOI | MR | Zbl

[10] Cerveau, D.; Mattei, J.-F. Formes intégrables holomorphes singulières, Astérisque, 97, Société Mathématique de France, Paris, 1982, 193 pages | Zbl

[11] Cerveau, D.; Sad, P. Liouvillian integration and Bernoulli foliations, Trans. Amer. Math. Soc., Volume 350 (1998) no. 8, pp. 3065-3081 | DOI | MR | Zbl

[12] Claudon, B.; Loray, F.; Pereira, J. V.; Touzet, F. Compact leaves of codimension one holomorphic foliations on projective manifolds (2015) (arXiv:1512.06623)

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

[14] Cousin, G. Connexions plates logarithmiques de rang deux sur le plan projectif complexe, IRMAR (2011) (PhD Thesis http://tel.archives-ouvertes.fr/tel-00779098)

[15] Cousin, G. Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI, Ann. Inst. Fourier (Grenoble), Volume 64 (2014) no. 2, pp. 699-737 http://aif.cedram.org/item?id=AIF_2014__64_2_699_0 | DOI | Numdam | MR | Zbl

[16] Cousin, G.; Pereira, J. V. Transversely affine foliations on projective manifolds, Math. Res. Lett., Volume 21 (2014) no. 5, pp. 985-1014 | DOI | MR | Zbl

[17] Deligne, P. Équations différentielles à points singuliers réguliers, Lect. Notes in Math., 163, Springer-Verlag, Berlin, 1970 | Zbl

[18] Godbillon, C. Feuilletages. Études géométriques, Progress in Math., 98, Birkhäuser Verlag, Basel, 1991, xiv+474 pages | Zbl

[19] Kedlaya, K. Good formal structures for flat meromorphic connections, I: surfaces, Duke Math. J., Volume 154 (2010) no. 2, pp. 343-418 | DOI | MR | Zbl

[20] Lazarsfeld, R. Positivity in algebraic geometry. I, Ergeb. Math. Grenzgeb. (3), 48, Springer-Verlag, Berlin, 2004, xviii+387 pages | DOI | MR

[21] Lins Neto, A. Construction of singular holomorphic vector fields and foliations in dimension two, J. Differential Geom., Volume 26 (1987) no. 1, pp. 1-31 http://projecteuclid.org/euclid.jdg/1214441174 | DOI | MR | Zbl

[22] Lins Neto, A. Some examples for the Poincaré and Painlevé problems, Ann. Sci. École Norm. Sup. (4), Volume 35 (2002) no. 2, pp. 231-266 | DOI | Numdam | Zbl

[23] Loray, F. Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux (2006) (hal-00016434)

[24] Loray, F.; Pereira, J. V. 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] Loray, F.; Pereira, J. V.; Touzet, F. Singular foliations with trivial canonical class (2011) (arXiv:1107.1538) | Zbl

[26] Mal’cev, A. I. On the faithful representation of infinite groups by matrices, Mat. Sb., Volume 8 (1940), pp. 405-422 English transl.: Amer. Math. Soc. Transl. (2) 45 (1965), p. 1–18

[27] Malgrange, B. Connexions méromorphes. II. Le réseau canonique, Invent. Math., Volume 124 (1996) no. 1-3, pp. 367-387 | DOI | Zbl

[28] Malgrange, B. On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, Volume 23 (2002) no. 2, pp. 219-226 | DOI | MR | Zbl

[29] Martinet, J.; Ramis, J.-P. Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Publ. Math. Inst. Hautes Études Sci. (1982) no. 55, pp. 63-164 | DOI | Numdam | Zbl

[30] 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

[31] Mochizuki, T. Good formal structure for meromorphic flat connections on smooth projective surfaces, Algebraic Analysis and Around (Advanced Studies in Pure Math.), Volume 54, Math. Soc. Japan, Tokyo, 2009, pp. 223-253 | MR | Zbl

[32] Moerdijk, I.; Mrčun, J. Introduction to foliations and Lie groupoids, Cambridge Studies in Adv. Math., 91, Cambridge University Press, Cambridge, 2003, x+173 pages | DOI | MR | Zbl

[33] Mumford, D. The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. Inst. Hautes Études Sci. (1961) no. 9, pp. 5-22 | DOI | Numdam | MR | Zbl

[34] Neeman, A. Ueda theory: theorems and problems, Mem. Amer. Math. Soc., 81, no.  415, American Mathematical Society, Providence, R.I., 1989, vi+123 pages | DOI | Zbl

[35] Pereira, J. V. Fibrations, divisors and transcendental leaves, J. Algebraic Geom., Volume 15 (2006) no. 1, pp. 87-110 | DOI | MR | Zbl

[36] Pereira, J. V.; Sad, P. Rigidity of fibrations, Differential equations and singularities. 60 years of J. M. Aroca (Astérisque), Volume 323, Société Mathématique de France, Paris, 2009, pp. 291-299 | Zbl

[37] Rousseau, E.; Touzet, F. Curves in Hilbert modular varieties (2015) (arXiv:1501.03261)

[38] Sabbah, C. Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension $2$, Astérisque, 263, Société Mathématique de France, Paris, 2000 | Zbl

[39] Scárdua, B. A. 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

[40] Totaro, B. The topology of smooth divisors and the arithmetic of abelian varieties, Michigan Math. J., Volume 48 (2000), pp. 611-624 | DOI | MR | Zbl

[41] Totaro, B. Moving codimension-one subvarieties over finite fields, Amer. J. Math., Volume 131 (2009) no. 6, pp. 1815-1833 | DOI | MR | Zbl

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