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 $\mathbf{C}$?



On delta complex structures of complex quasi-projective varieties 
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.


