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
Frank Loray 1; Jorge Vitório Pereira 2; Frédéric Touzet 1
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TY - JOUR AU - Frank Loray AU - Jorge Vitório Pereira AU - Frédéric Touzet 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 - https://jep.centre-mersenne.org/articles/10.5802/jep.34/ DO - 10.5802/jep.34 LA - en ID - JEP_2016__3__263_0 ER -
%0 Journal Article %A Frank Loray %A Jorge Vitório Pereira %A Frédéric Touzet %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 https://jep.centre-mersenne.org/articles/10.5802/jep.34/ %R 10.5802/jep.34 %G en %F JEP_2016__3__263_0
Frank Loray; Jorge Vitório Pereira; Frédéric Touzet. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.34/
[1] - “Structure des connexions méromorphes formelles de plusieurs variables et semi-continuité de l’irrégularité”, Invent. Math. 170 (2007) no. 1, p. 147-198 | DOI | MR | Zbl
[2] - “Characteristic varieties of quasi-projective manifolds and orbifolds”, Geom. Topol. 17 (2013) no. 1, p. 273-309 | DOI | MR | Zbl
[3] - “Sur l’intégration des équations différentielles holomorphes réduites en dimension deux”, Bol. Soc. Brasil. Mat. (N.S.) 30 (1999) no. 3, p. 247-286 | DOI | Zbl
[4] - “Minimal models of foliated algebraic surfaces”, Bull. Soc. math. France 127 (1999) no. 2, p. 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 | Zbl
[6] - “Holomorphic foliations with Liouvillian first integrals”, Ergodic Theory Dynam. Systems 21 (2001) no. 3, p. 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) 56 (2006) no. 3, p. 735-779 | DOI | Numdam | Zbl
[9] - “Complex codimension one singular foliations and Godbillon-Vey sequences”, Moscow Math. J. 7 (2007) no. 1, p. 21-54, 166 | DOI | MR | Zbl
[10] - Formes intégrables holomorphes singulières, Astérisque, vol. 97, Société Mathématique de France, Paris, 1982 | Zbl
[11] - “Liouvillian integration and Bernoulli foliations”, Trans. Amer. Math. Soc. 350 (1998) no. 8, p. 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. 144 (2008) no. 5, p. 1271-1331 | DOI | MR | Zbl
[14] - Connexions plates logarithmiques de rang deux sur le plan projectif complexe, IRMAR, 2011, PhD Thesis
[15] - “Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI”, Ann. Inst. Fourier (Grenoble) 64 (2014) no. 2, p. 699-737 | DOI | Numdam | MR | Zbl
[16] - “Transversely affine foliations on projective manifolds”, Math. Res. Lett. 21 (2014) no. 5, p. 985-1014 | DOI | MR | Zbl
[17] - Équations différentielles à points singuliers réguliers, Lect. Notes in Math., vol. 163, Springer-Verlag, Berlin, 1970 | Zbl
[18] - Feuilletages. Études géométriques, Progress in Math., vol. 98, Birkhäuser Verlag, Basel, 1991 | Zbl
[19] - “Good formal structures for flat meromorphic connections, I: surfaces”, Duke Math. J. 154 (2010) no. 2, p. 343-418 | DOI | MR | Zbl
[20] - Positivity in algebraic geometry. I, Ergeb. Math. Grenzgeb. (3), vol. 48, Springer-Verlag, Berlin, 2004 | DOI | MR
[21] - “Construction of singular holomorphic vector fields and foliations in dimension two”, J. Differential Geom. 26 (1987) no. 1, p. 1-31 | DOI | MR | Zbl
[22] - “Some examples for the Poincaré and Painlevé problems”, Ann. Sci. École Norm. Sup. (4) 35 (2002) no. 2, p. 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. 18 (2007) no. 6, p. 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. 8 (1940), p. 405-422, English transl.: Amer. Math. Soc. Transl. (2) 45 (1965), p. 1–18
[27] - “Connexions méromorphes. II. Le réseau canonique”, Invent. Math. 124 (1996) no. 1-3, p. 367-387 | DOI | Zbl
[28] - “On nonlinear differential Galois theory”, Chinese Ann. Math. Ser. B 23 (2002) no. 2, p. 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, p. 63-164 | DOI | Numdam | Zbl
[30] - “Hilbert modular foliations on the projective plane”, Comment. Math. Helv. 80 (2005) no. 2, p. 243-291 | DOI | MR | Zbl
[31] - “Good formal structure for meromorphic flat connections on smooth projective surfaces”, in Algebraic Analysis and Around, Advanced Studies in Pure Math., vol. 54, Math. Soc. Japan, Tokyo, 2009, p. 223-253 | MR | Zbl
[32] - Introduction to foliations and Lie groupoids, Cambridge Studies in Adv. Math., vol. 91, Cambridge University Press, Cambridge, 2003 | 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, p. 5-22 | DOI | Numdam | MR | Zbl
[34] - Ueda theory: theorems and problems, Mem. Amer. Math. Soc., vol. 81, no. 415, American Mathematical Society, Providence, R.I., 1989 | DOI | Zbl
[35] - “Fibrations, divisors and transcendental leaves”, J. Algebraic Geom. 15 (2006) no. 1, p. 87-110 | DOI | MR | Zbl
[36] - “Rigidity of fibrations”, in Differential equations and singularities. 60 years of J. M. Aroca, Astérisque, vol. 323, Société Mathématique de France, Paris, 2009, p. 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, vol. 263, Société Mathématique de France, Paris, 2000 | Zbl
[39] - “Transversely affine and transversely projective holomorphic foliations”, Ann. Sci. École Norm. Sup. (4) 30 (1997) no. 2, p. 169-204 | DOI | Numdam | MR | Zbl
[40] - “The topology of smooth divisors and the arithmetic of abelian varieties”, Michigan Math. J. 48 (2000), p. 611-624 | DOI | MR | Zbl
[41] - “Moving codimension-one subvarieties over finite fields”, Amer. J. Math. 131 (2009) no. 6, p. 1815-1833 | DOI | MR | Zbl
[42] - “Sur les feuilletages holomorphes transversalement projectifs”, Ann. Inst. Fourier (Grenoble) 53 (2003) no. 3, p. 815-846 | DOI | Numdam | MR | Zbl
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