Why is a complex algebraic variety a pseudomanifold (a filtered space which "filtered-locally" looks like a product of an open subset of real space and a cone of a compact filtered space)? Can anyone give a reference for this?

Here is what I understand. A complex algebraic variety $X$ admits a Whitney stratification. I do not really understand if in Whitney's original paper he assumes that the variety is embedded in a smooth manifold or not (see this question). But the 1976 paper by Verdier works in the general setting. Now, how do we get from a Whitney stratification to a pseudomanifold structure? Here is what I would do. Say, the $S_\alpha$ are the strata of a Whitney stratification. For any $i$ let $X^i$ be the union of the $i$-dimensional strata. Then we get a finite filtration of $X$ whose strata are precisely the $S_\alpha$. From what I understand, we need to assume that $X$ is pure of dimension $n$ and then define a new filtration $X \supseteq Y^{2n} \supseteq Y^{2n-1} \supseteq \cdots \supseteq$ with $Y^{2j} = Y^{2j+1} = X^j$ for all $j$. Why do we get a stratified pseudomanifold structure with respect to this filtration? This is claimed in Kirwan-Woolf "An Introduction to Intersection Homology Theory", Theorem 4.10.5 for $X$ quasi-projective with a reference to Borel's "Intersection Homology", more precisely the article by A'Campo in there. However, I do not understand what A'Campo is doing there and if the final claim (page 45) really shows that. Moreover, A'Campo works with an analytic space in general, so why restrict to quasi-projective?

So, I'd be happy already about an actual reference of the above claim. Is it always true that regardless of the pseudomanifold structure (in particular, regardless of the Whitney stratification) the union of the top-dimensional strata of a pseudomanifold structure on a variety is precisely the (algebro-geometric) smooth locus of the variety?


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