Construction et classification de certaines solutions algébriques des systèmes de Garnier

Abstract

22 pagesInternational audienceIn this paper, we classify all (complete) non elementary algebraic solutions of Garnier systems that can be constructed by Kitaev's method: they are deduced from isomonodromic deformations defined by pulling back a given fuchsian equation E by a family of ramified covers. We first introduce orbifold structures associated to a fuchsian equation. This allow to get a refined version of Riemann-Hurwitz formula and then to promtly deduce that E is hypergeometric. Then, we can bound exponents and degree of the pull-back maps and further list all possible ramification cases. This generalizes a result due to C. Doran for the Painleve VI case. We explicitely construct one of these solutions

A new two-parameter family of isomonodromic deformations over the five punctured sphere

“…where (p i , q i ) i are algebraic functions of t 1 , t 2 and H 1 , H 2 are explicit Hamiltonians given in Proposition 3.3 (see also [8,16]). More precisely if one sets S q := q 1 + q 2 , P q := q 1 q 2 , S t := t 1 + t 2 and P t := t 1 t 2 one has the following relations:…”

“…In this paragraph, we prove that our family of monodromy representations cannot be generically obtained through a pullback method [8,9] by showing that it does not factor through a curve [4]. 5.1.…”

Classification of algebraic solutions of irregular Garnier systems

“…Pull-back algebraic solutions. Another way to construct algebraic isomonodromic deformations (see [15,1,29,2,30,31,32,46,47,14,42]) is to fix a differential equation (C 0 , E 0 , ∇ 0 ) and consider an algebraic family φ t : C t → C 0 of ramified covers, where t ∈ P a projective variety. The pull-back (C t , φ * t E 0 , φ * t ∇ 0 ) provides an algebraic isomonodromic deformation.…”

“…Sections 6, 7, 8 and 9 are devoted to the classification of such solutions. Inspired by the similar classification in the logarithmic case established by the first author in [14], we define the irregular analogues of curve, Teichmüller and moduli spaces, Euler characteristic and Riemann-Hurwitz formula. Then we prove that, assuming E 0 irregular with differential Galois group not reduced to the diagonal or dihedral group (to avoid classical solutions), E 0 is of degenerate hypergeometric type (at most 3 poles counted with multiplicity) and the cover degree of φ t is bounded by 6.…”

Finite orbits of the pure braid group on the monodromy of the 2-variable Garnier system

“…In order to keep down the number of pages and of technical lemmata, we restrict our classification to exceptional orbits, namely orbits for which the corresponding monodromy group is not reducible, none of the monodromy matrices is a multiple of the identity and at most one projection giving either a Kitaev or a Picard orbit is allowed. Therefore, our classification does not include the solutions found by Tsuda [28] by calculating fixed points of bi-rational canonical transformations, nor the ones found by Diarra in [5] using the method of pull-back introduced in [6] and [1], nor the families of algebraic solutions obtained by Girand in [12] by restricting a logarithmic flat connection defined on the complement of a quintic curve on P 2 on generic lines of the projective plane -indeed these algebraic solutions have at least two projections giving Kitaev orbits.…”