# On delta complex structures of complex quasi-projective varieties

Q1.Given a quasi-projective variety $X$ over $\mathbf{C}$, is it always possible to find a $\Delta$-complex structure on $X$?

Q2. What is a good reference which gives a survey about what we know of $CW$-complex structures of quasi-projective varieties over $\mathbf{C}$?

-

There are very general triangulability results for real (semi)algebraic sets (sets cut out by inequalities of real polynomials), and even for semianalytic and subanalytic sets. Lojasiewicz has some papers from the 60s on semianalytic sets; Hironaka and Hardt also have papers on the subject; and the book Real Algebraic Geometry by Bochnak, Coste, and Roy treats at least the semialgebraic case.

-
So I found a proof in the real algebraic geometry book of the triangulability of a compact semi-algebraic set. Many thanks for this reference. –  Hugo Chapdelaine Jul 20 '11 at 18:55
Actually, any complex algebraic variety can be triangulated as a pseudomanifold, which is a simplicial complex such that, if the variety has real dimension $n$, then any simplex is contained in an $n$-simplex, any $(n-1)$-simplex is contained in exactly two $n$-simplexes, and any two $n$-simplexes can be connected by a sequence of $n$-simplexes such that two consecutive ones share exactly one $(n-1)$-face.