Asymptotics for general connections at infinity

Overview & Synopsis

For a standard path of connections going to a genericpoint at infinity in the moduli space $M_{DR}$ ofconnections on a compact Riemann surface, we show thatthe Laplace transform of the family of monodromy matriceshas an analytic continuation with locally finite branching.In particular the convex subset representing the exponentialgrowth rate of the monodromy is a polygon, whose verticesare in a subset of points described explicitly in terms of the spectral curve. Unfortunately we don't get anyinformation about the size of the singularities of the Laplace transform, which is why we can't get asymptoticexpansions for the monodromy.