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Contractions of Lie algebroids and the moduli space of flat connections.

Le : 08/10/2013 16h00
Par : Pietro Tortella (LAREMA, Université d’Angers)
Lieu : I 103
Lien web :
Résumé : The non abelian Hodge correspondence establishes a (smooth) isomorphism between the moduli spaces of stable flat connections and stable Higgs bundles over a smooth projetive complex variety. Both flat connections and Higgs bundles are representations of different Lie algebroid structures one can put on the tangent bundle (respectively, the canonical and the trivial one). Moreover, let us remark that the trivial Lie algebroid structure of the tangent bundle is a contraction of the canonical one. We construct a moduli space M_tau parametrizing semistable pairs (S, D), where S is a Lie algebroid structure of the tangent bundle and D is a representation of S. This space may be used to study the moduli space of flat connections and of Higgs bundles. For instance, Simpson's lambda connections are naturally included in this space, and many constructions can be generalized to this setting. In particular, there is an action of the space of automorphisms of the tangent bundle over M_tau that generalize the C^* action on the moduli space of lambda connections, and a quotient of M_tau by this action provides a (partial) compactification of the moduli space of flat connections.