Zeros of the Jimbo, Miwa, Ueno tau function

We introduce a family of local deformations for meromorphic connections on P1 in the neighborhood of a higher rank (simple) singularity. Following the scheme in Malgrange [Mathematique et Physique, Progress in Mathematics (Birkhäuser, Boston, 1983), Vol. 37, pp. 381–400, ibid., pp. 401–426; ibid., pp. 427–438] we use these local models to prove that the zeros of the tau function, introduced by Jimbo, Miwa, and Ueno in their pioneering work on “Birkhoff” deformations at irregular singular points [Physica D 2, 306–352; 2, 407–448 (1981); 4, 26–46 (1983); Publ. RIMS Kyoto Univ. 17-2, 703–721 (1981)], occur at precisely those points in the deformation space at which a certain Birkhoff–Riemann–Hilbert problem fails to have a solution.