---
date: 2022-09-05 14:04
modification date: Monday 5th September 2022 14:04:40
title: "Chow's theorem"
aliases: [Chow's theorem]
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- Tags: 
	- #AG/complex 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

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# Chow's theorem

- If $X \subseteq \PP^n$ is a closed analytic subspace, then $X$ is is a projective subscheme.
	- In general, for an algebraic scheme $(X, \OO_X)$ of ft over $\CC$, one can construct an analytic variety $(X^\an, \OO_{X}^\an)$ by essentially throwing away non-closed points and enriching the topology.
	- This yields a functor that induces an equivalence between projective schemes and compact analytic subschemes of $\PP^N$.
	- Idea why: a morphisms of schemes or analytic spaces is represented by its graph, which embeds in the product. So if the domain/range are both projective, so is the graph.
- $\CC$ has many holomorphic functions that are far from algebraic, so that $X$ is an analytic closed subset is necessary.