Abstract
Chow showed that every complex submanifold of ℙ n ℂ is an algebraic variety. The eventual goal of this chapter and the next is to outline the proof of a refined version of this due to Serre [101], usually referred to as “GAGA,” which is an acronym derived from the title of his paper. The first part of the theorem gives a correspondence between certain objects on ℙ n ℂ viewed as an algebraic variety and objects on ℙ n ℂ viewed as a complex manifold. These objects are coherent sheaves that are O-modules that are locally finitely presented in a suitable sense. Some of the formal properties of coherent sheaves are given here. Over affine and projective spaces there is a complete description of coherent sheaves in elementary algebraic terms, which makes this class particularly attractive. Chow’s theorem is recovered by applying GAGA to ideal sheaves, which are coherent.
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