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.
L’objet de cet article est d’établir un théorème de structure pour les feuilletages singuliers transversalement projectifs de codimension sur une variété projective lisse. Pour ce faire, nous étendons d’abord la classification de Corlette et Simpson de représentations de rang 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 . En particulier, nous montrons qu’une connexion méromorphe plate de rang avec des singularités irrégulières et des matrices de Stokes non triviales se factorise par une connexion sur une courbe.
Accepted:
Published online:
DOI: 10.5802/jep.34
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
@article{JEP_2016__3__263_0, author = {Loray, Frank and Pereira, Jorge Vit\'orio and Touzet, Fr\'ed\'eric}, title = {Representations of quasi-projective groups, flat connections and transversely~projective~foliations}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {263--308}, publisher = {ole polytechnique}, volume = {3}, year = {2016}, doi = {10.5802/jep.34}, mrnumber = {3522824}, zbl = {1353.37098}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jep.34/} }
TY - JOUR AU - Loray, Frank AU - Pereira, Jorge Vitório AU - Touzet, Frédéric TI - Representations of quasi-projective groups, flat connections and transversely projective foliations JO - Journal de l’École polytechnique — Mathématiques PY - 2016 SP - 263 EP - 308 VL - 3 PB - ole polytechnique UR - http://archive.numdam.org/articles/10.5802/jep.34/ DO - 10.5802/jep.34 LA - en ID - JEP_2016__3__263_0 ER -
%0 Journal Article %A Loray, Frank %A Pereira, Jorge Vitório %A Touzet, Frédéric %T Representations of quasi-projective groups, flat connections and transversely projective foliations %J Journal de l’École polytechnique — Mathématiques %D 2016 %P 263-308 %V 3 %I ole polytechnique %U http://archive.numdam.org/articles/10.5802/jep.34/ %R 10.5802/jep.34 %G en %F JEP_2016__3__263_0
Loray, Frank; Pereira, Jorge Vitório; Touzet, Frédéric. Representations of quasi-projective groups, flat connections and transversely projective foliations. Journal de l’École polytechnique — Mathématiques, Volume 3 (2016), pp. 263-308. doi : 10.5802/jep.34. http://archive.numdam.org/articles/10.5802/jep.34/
[1] 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] Characteristic varieties of quasi-projective manifolds and orbifolds, Geom. Topol., Volume 17 (2013) no. 1, pp. 273-309 | DOI | MR | Zbl
[3] 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] Minimal models of foliated algebraic surfaces, Bull. Soc. math. France, Volume 127 (1999) no. 2, pp. 289-305 | DOI | Numdam | MR | Zbl
[5] Birational geometry of foliations, Publicações Matemáticas do IMPA, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2004, iv+138 pages | Zbl
[6] 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] Théorie élémentaires des feuilletages holomorphes singuliers, Échelles, Belin, Paris, 2013
[8] 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] 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] Formes intégrables holomorphes singulières, Astérisque, 97, Société Mathématique de France, Paris, 1982, 193 pages | Zbl
[11] Liouvillian integration and Bernoulli foliations, Trans. Amer. Math. Soc., Volume 350 (1998) no. 8, pp. 3065-3081 | DOI | MR | Zbl
[12] Compact leaves of codimension one holomorphic foliations on projective manifolds (2015) (arXiv:1512.06623)
[13] 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] Connexions plates logarithmiques de rang deux sur le plan projectif complexe, IRMAR (2011) (PhD Thesis http://tel.archives-ouvertes.fr/tel-00779098)
[15] 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] Transversely affine foliations on projective manifolds, Math. Res. Lett., Volume 21 (2014) no. 5, pp. 985-1014 | DOI | MR | Zbl
[17] Équations différentielles à points singuliers réguliers, Lect. Notes in Math., 163, Springer-Verlag, Berlin, 1970 | Zbl
[18] Feuilletages. Études géométriques, Progress in Math., 98, Birkhäuser Verlag, Basel, 1991, xiv+474 pages | Zbl
[19] Good formal structures for flat meromorphic connections, I: surfaces, Duke Math. J., Volume 154 (2010) no. 2, pp. 343-418 | DOI | MR | Zbl
[20] Positivity in algebraic geometry. I, Ergeb. Math. Grenzgeb. (3), 48, Springer-Verlag, Berlin, 2004, xviii+387 pages | DOI | MR
[21] 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] 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] Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux (2006) (hal-00016434)
[24] 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] Singular foliations with trivial canonical class (2011) (arXiv:1107.1538) | Zbl
[26] 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] Connexions méromorphes. II. Le réseau canonique, Invent. Math., Volume 124 (1996) no. 1-3, pp. 367-387 | DOI | Zbl
[28] On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, Volume 23 (2002) no. 2, pp. 219-226 | DOI | MR | Zbl
[29] 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] Hilbert modular foliations on the projective plane, Comment. Math. Helv., Volume 80 (2005) no. 2, pp. 243-291 | DOI | MR | Zbl
[31] 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] Introduction to foliations and Lie groupoids, Cambridge Studies in Adv. Math., 91, Cambridge University Press, Cambridge, 2003, x+173 pages | DOI | MR | Zbl
[33] 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] 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] Fibrations, divisors and transcendental leaves, J. Algebraic Geom., Volume 15 (2006) no. 1, pp. 87-110 | DOI | MR | Zbl
[36] 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] Curves in Hilbert modular varieties (2015) (arXiv:1501.03261)
[38] Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension , Astérisque, 263, Société Mathématique de France, Paris, 2000 | Zbl
[39] 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] The topology of smooth divisors and the arithmetic of abelian varieties, Michigan Math. J., Volume 48 (2000), pp. 611-624 | DOI | MR | Zbl
[41] Moving codimension-one subvarieties over finite fields, Amer. J. Math., Volume 131 (2009) no. 6, pp. 1815-1833 | DOI | MR | Zbl
[42] 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
Cited by Sources: