Branched structures on Riemann surfaces
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- by Richard Mandelbaum PDF
- Trans. Amer. Math. Soc. 163 (1972), 261-275 Request permission
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.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 163 (1972), 261-275
- MSC: Primary 30.45
- DOI: https://doi.org/10.1090/S0002-9947-1972-0288253-1
- MathSciNet review: 0288253