Branched Projective Structures on a Riemann Surface and Logarithmic Connections

  • Indranil Biswas

    School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
  • Sorin Dumitrescu

    Université Côte d'Azur, Nice, France
  • Subhojoy Gupta

    Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
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Abstract

We study the set consisting of all branched holomorphic projective structures on a compact Riemann surface of genus and with a fixed branching divisor , where . Under the hypothesis that , for all , with a positive even integer such that , we show that coincides with a subset of the set of all logarithmic connections with singular locus , satisfying certain geometric conditions, on the rank two holomorphic jet bundle , where is a fixed holomorphic line bundle on such that . The space of all logarithmic connections of the above type is an affine space over the vector space of dimension . We conclude that is a subset of this affine space that has codimenison at a generic point.

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Indranil Biswas, Sorin Dumitrescu, Subhojoy Gupta, Branched Projective Structures on a Riemann Surface and Logarithmic Connections. Doc. Math. 24 (2019), pp. 2299–2337

DOI 10.4171/DM/726