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

Abstract

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 Complex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an [Formula: see text] linear system of ODEs with an irregular singularity of Poincaré rank 1 at [Formula: see text] and Fuchsian singularity at [Formula: see text], holomorphically depending on parameter [Formula: see text] within a polydisk in [Formula: see text] centered at [Formula: see text]. The eigenvalues of the leading matrix at [Formula: see text], which is diagonal, coalesce along a coalescence locus [Formula: see text] contained in the polydisk. Under minimal vanishing conditions on the residue matrix at [Formula: see text], we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808 .] that isomonodromic deformations can be extended to the whole polydisk, including [Formula: see text], 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 [Formula: see text], where it simplifies. Conversely, if the [Formula: see text]-dependent system is isomonodromic in a small domain contained in the polydisk not intersecting [Formula: see text], and if suitable entries of the Stokes matrices vanish, then [Formula: see text] is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlevé equations.