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In the book "Representation Theory and Complex Geometry” by Ginzburg-Chriss page 93 they claim that (the analytification of) every complex algebraic variety admits a closed embedding in a smooth manifold. They do not really provide a reference for this statement. Why is this true?

For quasi-projective varieties, the claim is fairly obvious. But it seems to me that they do not impose this restriction. Moreover, I would say that we need the variety to be separated for this but (I can live with that).

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  • $\begingroup$ I think the most widespread definition includes separatedness. At least it is so in Wikipedia (they cite Hartshorne) $\endgroup$ – მამუკა ჯიბლაძე May 27 '17 at 6:02
  • $\begingroup$ Yes, that's fine (I mean the claim would imply this anyways (my manifolds are Hausdorff at least)) $\endgroup$ – user110443 May 27 '17 at 6:06
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    $\begingroup$ Yeah, I know this, but it's just homotopy equivalent to a finite CW complex. $\endgroup$ – user110443 May 27 '17 at 7:49
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    $\begingroup$ There's a PhD thesis "Triangulation of locally semi-algebraic spaces" by Kyle Roger Hofman proving triangulability for any variety. Still, it's not a finite simplicial complex I guess, so I don't know how to get this into a manifold. $\endgroup$ – user110443 May 27 '17 at 9:38
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    $\begingroup$ Okay, what about this argument (can some expert confirm or correct this?): let $X$ be a separated variety. By Nagata compactification, it's an open dense in a proper variety $\hat{X}$. By the answer here, $\hat{X}$ admits a finite triangulation. Hence, $\hat{X}$ can be embedded in a smooth manifold $M$. Then $X$ is locally closed in $M$, so $X = U \cap Z$ for an open $U$ and a closed $Z$ in $M$. Hence, $X$ is closed in an open subset $U$ of $M$, and this is again a smooth manifold. $\endgroup$ – user110443 May 27 '17 at 11:13
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It took me ages to find this reference (and I actually found it in this post). There is a theorem by Acquistapace–Broglia–Tognoli in An embedding theorem for real analytic spaces stating:

Theorem. Let $X$ be a paracompact connected $n$-dimensional analytic space and suppose that $q := \sup_{x \in X} \dim T_x X < \infty$. Then $X$ admits a closed $C^\omega$-immersion into $\mathbb{R}^{n+q}$.

This is a generalization of Grauert's embedding theorem for real analytic manifolds. From this we obtain:

Corollary. Let $X$ be a separated complex algebraic variety. Then $X^{\mathrm{an}}$ admits a topological closed embedding into $\mathbb{R}^N$ for some $N$.

Proof. $X^{\mathrm{an}}$ is a paracompact complex (thus also real) analytic space with finitely many connected components and with bounded tangent space dimensions. Now, apply the theorem.

Remark 1. The only thing that surprises me is why the ABT paper has only one citation on MathSciNet since 1979; I don't know if there's another source for the above theorem (I hope it's right actually).

Remark 2. I don't know if there isn't a simpler argument.

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