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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 $\mathbf{C}$?

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Hugo Chapdelaine
    Commented Jul 20, 2011 at 18:55
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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.

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  • $\begingroup$ do you have a reference for this result? $\endgroup$
    – babubba
    Commented Jun 23, 2011 at 22:14
  • $\begingroup$ I don't remember exactly now, but Wikipedia suggests to look at Dieudonné History of Algebraic and Differential Topology. Probably you can find the result there, and a reference for the proof if not included. $\endgroup$
    – Fernando Muro
    Commented Jun 23, 2011 at 22:27
  • $\begingroup$ @Fernando, do you have a reference? So I looked quickly at Dieudonne's book and I found the definition of pseudomanifold but I could not find the result. $\endgroup$
    – Hugo Chapdelaine
    Commented Jun 24, 2011 at 21:01
  • $\begingroup$ @Hugo Have a look at the paper by Goresky and MacPherson on intersection homology then. $\endgroup$
    – Fernando Muro
    Commented Jun 25, 2011 at 21:56

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