Representations of quasi-projective groups, flat connections and transversely projective foliations
Journal de l’École polytechnique - Mathématiques, Volume 3 (2016), pp. 263-308.

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 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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.34
Classification: 37F75,  34M40,  32S40
Keywords: Foliation, transverse structure, birational geometry, flat connections, irregular singular points, Stokes matrices
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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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://www.numdam.org/articles/10.5802/jep.34/

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