Let $V$ be a complex quasi-projective variety, we know from H. Whitney's and B Teissier works on stratifications of algebraic varieties that $V$ has an intrinsic stratification $$X_0\subset X_2\subset\ldots\subset X_{2n}$$ made out by complex quasi-projective smooth varieties that satisfy Whitney conditions $a$ and $b$. This stratification induces a structure of topological stratified pseudomanifold on $V$ meaning in particular that any point $x\in X_i$ has a conical chart, a stratified homeomorphism: $$\phi:U_x\rightarrow V_x\times cL$$ where $L$ is a topological stratified pseudomanifold of dimension $2n-i-1$, $cL$ is the cone on $L$, $V_x$ is an open neighbourhood of $x\in X_i$ and $U_x$ is an open neighbourhood of $x\in V$.

A priori $\phi$ is just a stratified homeomorphism (for a proof one can look at N. A'Campo survey in Armand Borel et al. lecture notes "Intersection cohomology" chapter IV Birkhauser), but we can ask wether if this intrinsic stratification gives us a Piecewise Linear stratification.

In 1984 A'Campo explains that this question is open for analytic spaces, I was wondering if an answer is known in the case of complex algebraic varieties.

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    $\begingroup$ @OliverStraser: I'm not sure it's so straightforward. Goresky shows that there is a triangulation consistent with the stratification, but that's not obviously the same as the stratification being consistent with a PL stratified pseudomanifold structure. $\endgroup$ – Greg Friedman Oct 26 '15 at 6:14
  • $\begingroup$ However in the book of Markus Banagl a similiar argument is used: " By [Gor78], a Whitney stratified pseudomanifold X can be triangulated so that the Whitney stratification defines a PL stratification of X." $\endgroup$ – Oliver Straser Oct 26 '15 at 6:20

So i turned my comment into an answer after reading [1] again.

A Whitney stratification, i.e. a stratification satisfying Whitney's condition b (and so automaticly a), induces a triangulation compatible with the stratification. Especially this gives a PL-stratification compatible with the Whitney stratification:

See the comment: " By [Gor78], a Whitney stratified pseudomanifold X can be triangulated so that the Whitney stratification defines a PL stratification of X." in [2]

[1] Goresky, Mark; Triangulation of stratified objects. (Proc. Amer. Math. Soc. 72 (1978), 193-200) [2] Banagl, Markus; Topology of stratified spaces.

  • $\begingroup$ Thank you Oliver, I was a bit confused by N. A'Campo's comment (his text is from 1983-1984). Following your answer, I looked to Mark Goresky's note and in this paper he proved was I was searching for. This triangulation result was also proved by many people before the 1980's, for example in F. Johnson's PHD thesis (published in 1983: "On the triangulation of stratified sets and singular varieties". Trans. Amer. Math. Soc. 275 (1983), no. 1, 333–343). $\endgroup$ – David C Oct 26 '15 at 7:46
  • $\begingroup$ As above, I agree that Goresky (and others - Verona seems to be the final word in this area) have show that there are triangulations compatible with the stratifications. But I'd be interested in a reference or proof to show that this implies the existence of the PL conical charts, compatible with the given stratification, that David mentions in his question and that are part of the abstract definition of PL stratified pseudomanifold (which I think is what David is interested in). $\endgroup$ – Greg Friedman Oct 31 '15 at 7:55
  • $\begingroup$ Without such an argument, it's not clear to me that we have a "PL stratification" as opposed to just a PL space with triangulations compatible with the filtration. $\endgroup$ – Greg Friedman Oct 31 '15 at 7:55

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