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 ϕ G {\phi _G} , into the moduli spaces of flat G-bundles on M . ( G = GA ( 1 , C ) M.\;(G = {\text {GA}}(1,{\text {C}}) or PL ( 1 , C ) . ) {\text {PL}}(1,{\text {C}}).) Conditions are determined under which ϕ G {\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 ϕ G {\phi _G} is is fact holomorphic on each such variety.