Branched Structures and Affine and Projective Bundles on Riemann Surfaces

Abstract

A classification for analytic branched G-structures on a Riemann surface M is provided by means of a map ${\phi _G}$, into the moduli spaces of flat G-bundles on $M.\;(G = {\text {GA}}(1,{\text {C}})$ or ${\text {PL}}(1,{\text {C}}).)$ Conditions are determined under which ${\phi _G}$ 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 ${\phi _G}$ is is fact holomorphic on each such variety.