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By Chow's lemma, I mean any variant of the following basic result in algebraic geometry relating complete varieties to projective varieties:

Lemma. For any complete variety $X$, there exist a projective variety $\tilde{X}$ and a surjective birational map $\tilde{X} \to X$.

For example, Stacks gives this generalisation:

Lemma. For any noetherian scheme $S$ and any separated $S$-scheme $X$ of finite type, there exist an $S$-scheme $\tilde{X}$ and a surjective proper morphism $\pi : \tilde{X} \to X$ such that $\tilde{X}$ admits an immersion into $\mathbb{P}^n_S$ (for some $n$) and there is a dense open $U \subseteq X$ such that $\pi : \pi^{-1} U \to U$ is an isomorphism.

I am interested in the history of this result.

Question. What is the original version of Chow's lemma, when and where was it proved/published, and what was it used for?

In Weil's Foundations of algebraic geometry, the very definition of completeness seems to build in the core of the proof of Chow's lemma. The resemblance is enough to make me wonder whether Chow's lemma came first and Weil turned it upside-down to get the definition of completeness. Unfortunately I was not able to trace the result further back than EGA II, and I did not find anything resembling it in Chow's famous paper On compact complex analytic varieties either, so I am a bit at a loss.

Zhen Lin
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    The title and the question don't have matching spirit. Do you want to know what Chow's lemma really 'means' (given the benefit of modern terminology/insight), or do you really want to know about its historical origin? – Alex Youcis Sep 14 '22 at 02:04
  • Interesting point. I think I would like to know the historical origin first. Using modern definitions sometimes makes historical results a triviality – at least if you only consider the statements and ignore the proofs. – Zhen Lin Sep 14 '22 at 03:50

1 Answers1

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Milne [Algebraic geometry] cites Serre [1956, Géométrie algébrique et géométrie analytique, Proposition 6] and Chow [1957, On the projective embedding of homogeneous varieties, Lemma 1] for the result. Serre also cites Chow and formulates the result thus:

pour toute variété algébrique $X$, il existe une variété projective $Y$, une partie $U$ de $Y$, Z-ouverte et Z-dense dans $Y$, et une application régulière surjective $f : U \to X$ dont le graphe $T$ soit Z-fermé dans $X \times Y$. On a $U = Y$ si et seulement si $X$ est complète.

(Here, « Z-ouverte » and « Z-dense » refer to the Zariski topology, as opposed to the complex-analytic topology.)

On the other hand, Chow writes:

Let $V$ be a variety and let $k$ be a field of definition for $V$; then there exist a complete normal subvariety $V'$ in a projective space and a birational correspondence $F$ between $V'$ and $V$, both defined over a purely inseparable extension $k'$ of $k$, such that $F$ is defined at every point in a $k'$-open subset $W$ in $V'$ and only at such points, and the image of $W$ under $F$ is $V$; furthermore, the inverse image of every point in $V$ under $F$ is a closed subset in $V'$.

It seems Chow used the lemma in order to extend results about projective varieties to complete varieties. Anyway, my speculation that Weil's definition of completeness comes from Chow's lemma appears to be incorrect.

Zhen Lin
  • 97,105