The Jacobian Variety

  • C. Herbert Clemens
Part of the The University Series in Mathematics book series (USMA)


Again in this chapter we will assume a certain vague familiarity with the cohomology of sheaves and the theory of line bundles. This material is readily accessible in the modern literature, and some of it can be guessed once one is familiar with Čech cohomology of topological spaces. Our goal in this chapter is to examine in detail the Jacobian variety which is associated in an intrinsic way with each compact complex manifold of dimension 1. It is the existence of this auxiliary variety that makes the theory of compact manifolds of dimension 1 so much more beautiful and complete than the theory of complex manifolds of higher dimensions. The higher-dimensional theory often still consists of a struggle to resurrect or replace in some special case or other the one-dimensional theory.


Line Bundle Theta Function Chern Class Holomorphic Line Bundle Compact Complex Manifold 
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  1. An algebraic proof of this fact has been given in recent years by David Mumford in “Theta Characteristics of an Algebraic Curve,” Annales scientifiques de l’Ecole Normale Supérieure,vol. 4, 1971, pp. 181–192.Google Scholar
  2. J. Lewittes, “Riemann Surfaces and the Theta Function,” Acta Math. vol. III, 1964, pp. 51–55.MathSciNetGoogle Scholar
  3. K. Kodaira and D. Spencer, “On Deformations of Complex Analytic Structures,” Annals of Mathematics, vol. 67, 1958, p. 328ff.MathSciNetMATHCrossRefGoogle Scholar
  4. A. Andreotti and A. Mayer, “On Period Relations for Abelian Integrals on Algebraic Curves,” Ann. Scuola Norm. Sup., Pisa, vol. 21, 1967, p. 209.MathSciNetGoogle Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • C. Herbert Clemens
    • 1
  1. 1.University of UtahSalt Lake CityUSA

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