The Jacobian Variety
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.
KeywordsLine Bundle Theta Function Chern Class Holomorphic Line Bundle Compact Complex Manifold
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- 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