The Narasimhan—Seshadri theorem for parabolic bundles: an orbifold approach

Abstract

Given a stable parabolic bundle over a Riemann surface, we study the problem of finding a compatible Yang-Mills connexion. When the parabolic weights are rational there is an equivalent problem on an orbifold bundle. When the weights are irrational our method is to choose a sequence of approximating rational weights, obtain a corresponding sequence of Yang-Mills connexions on the resulting orbifold bundles and obtain the solution as the limit of this sequence: we need to consider mildly singular connexions which locally about a marked point take the form d — Aid# + a . Here A is a constant diagonal matrix whose entries depend on the weights and their rational approximations, 0 = arg(z) for z a local uniformizing (orbifold) coordinate centred on the marked point and a is an L 2 1 connexion matrix. In this context we find all the necessary gauge-theoretic tools to prove the theorem, including a version of Uhlenbeck’s weak compactness theorem, provided | A| is sufficiently small. (One of the advantages of this approach is that we do analysis on a compact orbifold rather than on the punctured surface.) Our methods also allow us to consider the analogous problem for stable parabolic Higgs bundles.