# Stratification of complex algebraic varieties

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.

• @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. – Greg Friedman Oct 26 '15 at 6:14
• 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." – Oliver Straser Oct 26 '15 at 6:20