Families of compact Riemann surfaces which do not admit n th roots

Abstract

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 SurfaceLine BundleComplex ManifoldMapping Class GroupCompact Riemann SurfaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.