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

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 Malgrange, Jimbo, Miwa, and Ueno, in the non-degenerate case, there exists a universal integrable deformation inducing (via a unique map) all other deformations [M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformations of linear ordinary differential equations with rational coefficients I, Physica D 2 (1981) 306–352; B. Malgrange, Déformations de systèmes différentiels et microdifférentiels, in Séminaire E.N.S. Mathématique et Physique, eds. L. Boutet de Monvel, A. Douady and J.-L. Verdier, Progress in Mathematics, Vol. 37 (Birkhäuser, Basel, 1983), pp. 351–379; B. Malgrange, Sur les déformations isomonodromiques, II, in Séminaire E.N.S. Mathématique et Physique, eds. L. Boutet de Monvel, A. Douady and J.-L. Verdier, Progress in Mathematics, Vol. 37 (Birkhäuser, Basel, 1983), pp. 427–438; B. Malgrange, Deformations of differential systems, II, J. Ramanujan Math. Soc. 1 (1986) 3–15]. On the other hand, in the degenerate case, Sabbah proved, under sharp conditions, the existence of an integrable deformation of solutions, sharing many properties of the one constructed by Malgrange–Jimbo–Miwa–Ueno [C. Sabbah, Integrable deformations and degenerations of some irregular singularities, Publ. RIMS Kyoto Univ. 57(3–4) (2021) 755–794; arXiv:1711.08514v3]. 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 Thijsse and Wasow [Ph. G. A. Thijsse, Global holomorphic similarity to a Jordan form, Results Math. 8 (1985) 78–87; W. Wasow, Linear Turning Point Theory (Springer-Verlag, New York, 1985)]. [ABSTRACT FROM AUTHOR]