Branched Structures on Riemann Surfaces

Following results of Gunning on geometric realizations of projective structures on Riemann surfaces, we investigate more fully certain generalizations of such structures.We define the notion of a branched analytic cover on a Riemann surface M (of genus g) and specialize this to the case of branched projective and affine structures.Establishing a correspondence between branched projective and affine structures on M and the classical projective and affine connections on M we show that if a certain linear homogeneous differential equation involving the connection has only meromorphic solutions on M then the connection corresponds to a branched structure on M. Utilizing this fact we then determine classes of positive divisors on M such that for each divisor X) in the appropriate class the branched structures having as their branch locus divisor form a nonempty affine variety.Finally we apply some of these results to study the structures on a fixed Riemann surface of genus 2.