Families of compact Riemann surfaces which do not admit nth roots
Let π : V → B 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.)
KeywordsRiemann Surface Line Bundle Complex Manifold Mapping Class Group Compact Riemann Surface
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