Isomonodromic deformations of systems of linear differential equations with irregular singularities (Q2914410)

The author focuses on the following family of systems of \(d\) linear differential equations NEWLINE\[NEWLINE\frac{d y}{d z}=A(z, t)\,y,\qquad A(z, t)=\sum_{j=1}^n\,\sum_{k=1}^{r_j+1}\,\frac{A^j_{-k}(t)}{(z-a_j)^k},\qquad \sum_{j=1}^n\,A^j_{-1}(t)=0\, NEWLINE\]NEWLINE with the matrices \(A^j_{-k}(t)\) depending holomorphically on \(t\in D(t^0)\), where \(D(t^0)\) is a neighbourhood of a point \(t^0\) in the parameter space and minimal strictly positive Poincaré rank at the singular points \(a_1, \ldots, a_n\) equal to \(r_1, \ldots, r_n\), respectively.NEWLINENEWLINEThe author proves that this family of systems defines an isomonodromic deformation with parameters of deformations \,\(t=(a_1, a_2, \ldots , a_n)\) \,if and only if there exists a matrix-valued 1-form \(\omega(z, t)\) uniform on \(\mathbb{CP}^1 \times D(t^0)\;\setminus\bigcup_{i=1}^n \{z-a_i=0\}\) and such that \(\omega(z, t)=A(z, t)\,d z\) for each fixed \(t\in D(t^0)\) and \(d\, \omega(z, t)=\omega(z, t) \wedge \omega(z, t)\).NEWLINENEWLINEThis result is an extension of the classical results of \textit{L. Schlesinger} [``Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten'', J. für Math. 141, 96--145 (1912; JFM 43.0385.01)] and \textit{A. A. Bolibruch} [``Differential equations with meromorphic coefficients'', Proc. Steklov Inst. Math. 272, 13--43 (2011)] on the deformations of Fuchsian systems and of \textit{M. Jimbo, T. Miwa} and \textit{K. Ueno} [Physica D 2, No. 2, 306--352 (1981; Zbl 1194.34167)] on the deformations of systems with non-resonant irregular singularities.