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Construction de déformations isomonodromiques par revêtements

Abstract : Garnier system of rank N is a system of nonlinear differential equations. Local solutions of dimension N, parameterize the isomonodromic deformations of scalar differential equations of order 2 on the Riemann sphere with 2N + 3 Fuchsian singularities (N + 3 essential singular points and N singularities apparent). These solutions are in general very transcendent, but it also has algebraic solutions. They appear for example when a deformed scalar equation in finite monodromy, or for certain reducible monodromy. One can also construct algebraic isomonodromic deformations by pulling back a Fuchsian equation determined by a family of N parameters branched coverings : the method used by Kitaev in the case N = 1, ie for the equation of Painlevé VI. We classify all algebraic solutions obtained by this method for arbitrary N, the monodromy is not elementary (in particular irreducible and infinite). There is not for N greater than or equal to 4. Some of these solutions are calculated explicitly in the final section. Kitaev's method allows the construction of incomplete algebraic solutions for any N (ie of dimension smaller than N, the solution is not necessarily algebraic) and also any genus. In the case of holomorphic connections of rank 2 on curves of genus 2, we classify non-elementary algebra deformations obtained by this method : they are all incomplete in one dimension. Also in this context, we study a family of four dimension 2-parameter deformations obtained from solutions of Garnier systems of rank N = 2. This family, which appears on bi-elliptic curves, is characterized in terms of monodromy.
Mots-clés : espaces analytiques
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  • HAL Id : tel-00746795, version 1

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Karamoko Diarra. Construction de déformations isomonodromiques par revêtements. Equations aux dérivées partielles [math.AP]. Université Rennes 1, 2011. Français. ⟨tel-00746795⟩

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