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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Branched structures on Riemann surfaces

Author: Richard Mandelbaum
Journal: Trans. Amer. Math. Soc. 163 (1972), 261-275
MSC: Primary 30.45
MathSciNet review: 0288253
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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.

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Keywords: Branched structures, Riemann surfaces, affine structure, projective structure, affine connection, projective connection, divisor, affine variety, quadratic differentials, abelian differentials, Weierstrass points
Article copyright: © Copyright 1972 American Mathematical Society

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