The wild monodromy of the Fifth Painlevé equation and its action on wild character variety: an approach of confluence

Abstract

The article presents a confluence approach to the study of the fifth Painlevé equation and of the nonlinear Stokes phenomenon at its irregular singularity at infinity. First the relation between the nonlinear monodromy of Painlevé VI and the nonlinear Stokes phenomenon in Painlevé V is explained in detail. Then the (wild) character variety of the isomonodromic problem for Painlevé V is constructed through confluence from the character variety of the corresponding isomonodromic problem for Painlevé VI and a birational transformation between them is provided. The known description of the action of the nonlinear monodromy of Painlevé VI on its character variety is then used to describe the action of the nonlinear wild monodromy of Painlevé V on its character variety, that is the action of the group generated by the nonlinear monodromy, the nonlinear Stokes operators and the nonlinear exponential torus.

The space of monodromy data for the Jimbo-Sakai family of $q$-difference equations

“…This condition can be translated into an algebraic condition on a, cf. [32], [42] page 22, [47]; we have :…”

“…6 More precisely if t remains in an open disc of the 3-punctured sphere P 1 (C) \ {0, 1, ∞}. 7 We used the excellent presentation of [42].…”

“…The fibers 42 Π −1 (0), Π −1 (∞) ⊂ F have special significance. For example, they contain all reducible matrices.…”

“…only C * -action on (m 2,1 , m 2,2 ) is allowed. So we shall identify the image of f 1 = 0 in F to a subset of 42 We shall find in 6.3 that there are two more special fibers.…”

“…• each line of Σ belongs exactly to 5 tri-tangent planes. 50 They are due to Martin Klimes, Emmanuel Paul and the second author, and they are studied in a work in progress on the confluence of the Painlevé equations [42,43].…”