On the universality of integrable deformations of solutions of degenerate Riemann-Hilbert-Birkhoff problems

TL;DR

This paper proves a relative universality property for integrable deformations of solutions to degenerate Riemann-Hilbert-Birkhoff problems, extending known results and characterizing holomorphic matrix maps in several complex variables.

Contribution

It establishes a maximal class of integrable deformations induced by Sabbah's deformation, generalizing universality concepts to degenerate cases and multiple complex variables.

Findings

Existence and uniqueness of a maximal class of integrable deformations.

Sabbah's deformation satisfies a relative universal property.

Characterization of holomorphic matrix maps as locally holomorphically Jordanizable.

Abstract

This paper addresses the classification problem of integrable deformations of solutions of "degenerate" Riemann-Hilbert-Birkhoff (RHB) problems. These consist of those RHB problems whose initial datum has diagonal pole part with coalescing eigenvalues. On the one hand, according to theorems of B.Malgrange, M.Jimbo, T.Miwa, and K.Ueno, in the non-degenerate case, there exists a universal integrable deformation inducing (via a unique map) all other deformations. On the other hand, in the degenerate case, C.Sabbah proved (arXiv:1711.08514v4), under sharp conditions, the existence of an integrable deformation of solutions, sharing many properties of the one constructed by Malgrange-Jimbo-Miwa-Ueno. Albeit the integrable deformation constructed by Sabbah is not, stricto sensu, universal, we prove that it satisfies a $relative$ universal property. We show the existence and uniqueness of a maximal class of integrable deformations all induced (via a unique map) by Sabbah's integrable deformation. Furthermore, we show that such a class is large enough to include all generic integrable deformations whose pole and deformation parts are locally holomorphically diagonalizable. In itinere, we also obtain a characterization of holomorphic matrix-valued maps which are locally holomorphically Jordanizable. This extends, to the case of several complex variables, already known results independently obtained by Ph.G.A.Thijsse and W.Wasow.