Shatalov‐Sternin's construction of complex WKB solutions and the choice of integration paths

We re‐examine Shatalov‐Sternin's proof of existence of resurgent solutions of a linear ODE. In particular, we take a closer look at the “Riemann surface” (actually, a two‐dimensional complex manifold) whose existence, endless continuability and other properties are claimed by those authors. We present a detailed argument for a part of the “Riemann surface” relevant for the exact WKB method.