Representations of quasi-projective groups, flat connections and transversely projective foliations
[Représentations de groupes quasi-projectifs, connexions plates et feuilletages transversalement projectifs]
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.

Reçu le : 2015-10-16
Accepté le : 2016-07-03
Publié le : 2016-07-11
DOI : https://doi.org/10.5802/jep.34
Classification : 37F75,  34M40,  32S40
Mots clés: Feuilletage, structure transverse, géométrie birationnelle, connexion plate, points singuliers irréguliers, matrices de Stokes
@article{JEP_2016__3__263_0,
     author = {Frank Loray and Jorge Vit\'orio Pereira and Fr\'ed\'eric Touzet},
     title = {Representations of quasi-projective groups, flat connections and transversely~projective~foliations},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     pages = {263--308},
     publisher = {ole polytechnique},
     volume = {3},
     year = {2016},
     doi = {10.5802/jep.34},
     zbl = {1353.37098},
     mrnumber = {3522824},
     language = {en},
     url = {jep.centre-mersenne.org/item/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, Tome 3 (2016) , pp. 263-308. doi : 10.5802/jep.34. https://jep.centre-mersenne.org/item/JEP_2016__3__263_0/

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