Branched structures on Riemann surfaces

Abstract:

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 $\mathfrak {D}$ in the appropriate class the branched structures having $\mathfrak {D}$ 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.