ISOMONODROMY DEFORMATIONS AT AN IRREGULAR SINGULARITY WITH COALESCING EIGENVALUES

摘要

We consider an n x n linear system of ODEs with an irregular singularity of Poincare rank 1 at z = infinity, holomorphically depending on parameter t within a polydisk in C-n centered at t = 0, such that the eigenvalues of the leading matrix at z = infinity coalesce along a locus Delta contained in the polydisk, passing through t = 0. Namely, z = infinity is a resonant irregular singularity for t is an element of Delta. We analyze the case when the leading matrix remains diagonalizable at Delta. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon, and monodromy data as t varies in the polydisk, and their limits for t tending to points of Delta. When the system also has a Fuchsian singularity at z = 0, we show, under minimal vanishing conditions on the residue matrix at z = 0, that isomonodromic deformations can be extended to the whole polydisk (including Delta) 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 the fixed coalescence point t = 0. Conversely, when the system is isomonodromic in a small domain not intersecting Delta inside the polydisk, we give certain vanishing conditions on some entries of the Stokes matrices, ensuring that Delta is not a branching locus for the t-continuation of fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius manifolds is explained. An application to Painleve equations is discussed.