Déformations isomonodromiques des connexions de rang 2 sur les courbes

Abstract : We consider irreducible tracefree non-singular or meromorphic rank 2 connections over compact Riemann surfaces of arbitrary genus.
By deforming the curve, the position of the poles and the connection, we construct the universal isomonodromic deformation of such a connection. Our construction, which is specific to the tracefree rank 2 case, does not need any Stokes analysis for irregular singularities. It is thereby more elementary than the construction in arbitrary rank due to B. Malgrange and I. Krichever and it includes the case of resonant singularities in a natural way.
We prove that the underlying vector bundle is generically maximally stable along the universal isomonodromic deformation, provided that the initial connection is irreducible. For that purpose, we use an analytic version of M. Maruyama's semicontinuity-result and we explain the problem geometrically in terms of transversality in foliations. By means of explicit examples we show that the irreducibility-condition is necessary and that the analytic set of parameters which are non-generic in the above sens can be non-algebraic.
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Submitted on : Monday, February 2, 2009 - 4:05:02 PM
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Viktoria Heu. Déformations isomonodromiques des connexions de rang 2 sur les courbes. Mathématiques [math]. Université Rennes 1, 2008. Français. ⟨tel-00358039⟩

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