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Riemann Surfaces and Generalized Theta Functions

  • Book
  • © 1976

Overview

Authors:
  1. Robert C. Gunning

    1. Dept. of Mathematics, Princeton University, Princeton, USA

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Part of the book series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge (MATHE2, volume 91)

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Table of contents (4 chapters)

  1. Front Matter

    Pages I-XII
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  2. Complex Manifolds and Vector Bundles

    • Robert C. Gunning
    Pages 1-16

  3. Riemann Surfaces

    • Robert C. Gunning
    Pages 17-38

  4. Generalized Theta Functions

    • Robert C. Gunning
    Pages 39-90

  5. Prym Differentials

    • Robert C. Gunning
    Pages 91-129

  6. Back Matter

    Pages 130-168
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Keywords

About this book

The investigation of the relationships between compact Riemann surfaces (al­ gebraic curves) and their associated complex tori (Jacobi varieties) has long been basic to the study both of Riemann surfaces and of complex tori. A Riemann surface is naturally imbedded as an analytic submanifold in its associated torus; and various spaces of linear equivalence elasses of divisors on the surface (or equivalently spaces of analytic equivalence elasses of complex line bundies over the surface), elassified according to the dimensions of the associated linear series (or the dimensions of the spaces of analytic cross-sections), are naturally realized as analytic subvarieties of the associated torus. One of the most fruitful of the elassical approaches to this investigation has been by way of theta functions. The space of linear equivalence elasses of positive divisors of order g -1 on a compact connected Riemann surface M of genus g is realized by an irreducible (g -1)-dimensional analytic subvariety, an irreducible hypersurface, of the associated g-dimensional complex torus J(M); this hyper­ 1 surface W- r;;;, J(M) is the image of the natural mapping Mg- -+J(M), and is g 1 1 birationally equivalent to the (g -1)-fold symmetric product Mg- jSg-l of the Riemann surface M.

Authors and Affiliations

  • Dept. of Mathematics, Princeton University, Princeton, USA

    Robert C. Gunning

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