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

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

doi:10.5802/jep.34

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

Loray, Frank ¹ ; Pereira, Jorge Vitório ² ; Touzet, Frédéric ¹

¹ IRMAR, Université de Rennes 1 Campus de Beaulieu, 35042 Rennes Cedex, France
² IMPA Estrada Dona Castorina, 110, Horto, Rio de Janeiro, Brasil

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

Résumé
Abstract

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

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/jep.34

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

Affiliations des auteurs :

Loray, Frank ¹ ; Pereira, Jorge Vitório ² ; Touzet, Frédéric ¹

¹ IRMAR, Université de Rennes 1 Campus de Beaulieu, 35042 Rennes Cedex, France
² IMPA Estrada Dona Castorina, 110, Horto, Rio de Janeiro, Brasil

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     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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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. http://archive.numdam.org/articles/10.5802/jep.34/

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