Results on the extension of isomonodromy deformations to the case of a resonant irregular singularity

We explain some results of [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.], discussed in our talk [G. Cotti, Monodromy of semisimple Frobenius coalescent structures, in Int. Workshop Asymptotic and Computational Aspects of Com- plex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an n × n linear system of ODEs with an irregular singularity of Poincar ́e rank 1 at z = ∞ and Fuchsian singularity at z = 0, holomorphically depending on parameter t within a polydisk in Cn centered at t = 0. The eigenvalues of the leading matrix at ∞, which is diagonal, coalesce along a coalescence locus ∆ contained in the polydisk. Under minimal vanishing conditions on the residue matrix at z = 0, we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singular- ity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.] that isomonodromic deformations can be extended to the whole polydisk, including ∆, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at point of ∆, where it simplifies. Conversely, if the t-dependent system is isomonodromic in a small domain contained in the polydisk not intersecting ∆, and if suitable entries of the Stokes matrices vanish, then ∆ is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlev ́e equations.