  We study the WKB expansion of $2\times 2$ system of linear differential
equations with four fuchsian singularities. The main focus is on the generating
function of the monodromy symplectomorphism which, according to a recent paper
is closely related to the Jimbo-Miwa tau-function. We compute the first three
terms of the WKB expansion of the generating function and establish the link to
the Bergman tau-function.
