Asymptotic Geometry of the Moduli Space of Rank Two Irregular Higgs Bundles over the Projective Line

Abstract

We study the asymptotic behavior of Hitchin's hyperk\"ahler metric on the moduli space of rank two irregular Higgs bundles over $\mathbb{C}P^1$. Along a generic curve, we prove that the Hitchin metric is asymptotic to the semiflat metric at an arbitrary polynomial order. When there are no weakly parabolic singularities, the rate is exponential. In the case of four-dimensional moduli spaces, we prove that the semiflat metric is asymptotic to an ALG/ALG$^\ast$ model metric.