Branched Structures and Affine and Projective Bundles on Riemann Surfaces

A classification for analytic branched G-structures on a Riemann surface M is provided by means of a map φ G, into the moduli spaces of flat G-bundles on M.\;(G = GA(1,C) or PL(1,C).) Conditions are determined under which φ G is injective and these conditions are related to the total branching order of the G-structures. A decomposition of the space of analytic branched G-structures into a disjoint union of analytic varieties is exhibited and it is shown that φ G is is fact holomorphic on each such variety.