David Mumford gives in his book Algebraic Geometry I, Complex Projective Varieties on page 61 a definition of Chow's Lemma which has at least for me not a usual form:
If says that a closed $^*$-analytic subset $X \subset \mathbb{P}_{\mathbb{C}}^n$ is a finite union of algebraic varieties.
On 'closed $^*$-analytic subset': on page 61 Mumford defined a locally closed subset $X \subset \mathbb{C}^n$ as follows. Let $U \subset \mathbb{C}^n$ open set and $X \subset U$ closed in $U$. Then $X$ is $^*$-analytic if $X$ can be decomposed as
$$X = X^{(r)} \cup X^{(r-1)} \cup ... \cup X^{(0)} $$
where for all $i$, $ X^{(i)}$ is a $i$-dimensional complex submanifold of $U$ and the closure $\overline{X^{(i)}} \subset X^{(i)} \subset X^{(i-1)} \subset ... \subset X^{(0)}$.
Now in modern literature the Chow's Lemma is known as (this one is from Wikipedia but quoted from Grothendieck's EGA II):
If $X$ is a scheme that is proper over a noetherian base $ S $, then there exists a projective $S$ -scheme $ X' $ and a surjective $S$-morphism $ f:X'\to X $ that induces an isomorphism $ f^{{-1}}(U)\simeq U$ for some dense $U\subseteq X$.
Question: is there any relation between Mumford's version and Wikipedia version. Remark: the wikipedia version is exactly that one from EGA II.
My conjecture: if $S= \operatorname{Spec} (\mathbb{C})$, ie $X$ is a complex scheme or say for sake of simplicity a variety, are these two statements equivalent? For example I don't know but probably the wikipedia version could be proved by applying Mumford's version which allows using analytic methods.
But that's only a conjecture of mine, nothing more. Does anybody know more on this issue?
