Families of compact Riemann surfaces which do not admit nth roots

  • Patricia L. Sipe
Conference paper
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 11)


Let π : VB be a holomorphic family of compact Riemann surfaces of genus p ≥ 2 (to be defined in section 1). For any t ? B, the fiber X t = π-1(t) is a closed Riemann surface; the canonical line bundle K(X t ) is the holomorphic cotangent bundle of X t . A standard construction (see section 1 for details) produces a line bundle K rel(V) → V, called the relative canonical bundle, whose restriction to each Riemann surface X t V is equivalent to the canonical bundle K(X t ). (Throughout this paper, all line bundles will be holomorphic complex line bundles and equivalence will be holomorphic equivalence.)


Riemann Surface Line Bundle Complex Manifold Mapping Class Group Compact Riemann Surface 
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    Farkas, H. and Kra, I., “Riemann Surfaces,” Springer-Verlag, New York, 1980.MATHGoogle Scholar
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    Sipe, P.L., Roots of the canonical bundle of the universal Teichmuller curve and certain subgroups of the mapping class groups Math. Ann. 260 (1982), 67–92.MathSciNetMATHCrossRefGoogle Scholar
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    Sipe, P.L., Some finite quotients of the mapping class group of a surface ,Proceedings of the American Mathematical Society 97 (1986), 515–524.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Patricia L. Sipe
    • 1
  1. 1.Department of MathematicsSmith CollegeNorthamptonUSA

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