1. In the beginning Riemann created the surfaces. The periods of integrals of abelian differentials on a compact surface of genus g immediately attach a g-dimensional complex torus to X. If g ≥ 2, the moduli space of X depends on 3g — 3 complex parameters. Thus problems in one complex variable lead, from the very beginning, to studies in several complex variables. Complex tori and moduli spaces are complex manifolds, i.e. Hausdorff spaces with local complex coordinates z1,..., zn; holomorphic functions are, locally, those functions which are holomorphic in these coordinates.
KeywordsModulus Space Complex Space Complex Manifold Cohomology Group Coherent Sheave
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