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A construction that I've seen at least a couple times in Deligne's work is that any finite simplicial complex has the homotopy type of a pretty natural algebraic variety. This appears, for instance, in 6. of his 1974 ICM address and in his paper with Sullivan on vanishing of Chern classes of flat bundles. (So for instance, the construction is robust enough to "transfer" a topological vector bundle on the simplicial complex to an algebraic vector bundle on the variety.)

Unfortunately I don't really understand his construction and was hoping somebody could explain it to me.

My French is awful, but here's my attempt to translate what he says. Suppose you have a finite set S and a set $\mathcal{S}$ of subsets of S, corresponding to simplices, with the usual property that it's closed under subsets. Consider the space $\mathbb{R}^S$, and for $\sigma \subset S$, let $|\sigma|$ be the subspace spanned by the basis vector corresponding to $s \in \sigma$. Let

$$|S| = \bigcup_{\sigma \in \mathcal{S}} |\sigma|$$

Then the corresponding algebraic variety is the same thing construction $\mathbb{C}^S$.

I don't understand how to go from this to the desired result. Isn't $|S|$ obviously contractible (onto the origin)?

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    $\begingroup$ UIAM, one takes for $|\sigma|$ the affine subspace generated by the basis vectors, so that it does not contain the origin. $\endgroup$
    – ACL
    Commented Feb 28, 2017 at 22:06
  • $\begingroup$ Oh I see, that does make a lot more sense. $\endgroup$
    – user84144
    Commented Feb 28, 2017 at 22:14
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    $\begingroup$ IIRC, such a construction is described in springer.com/gp/book/9783540064329 $\endgroup$
    – Dima Pasechnik
    Commented Mar 1, 2017 at 12:55

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the geometric picture is

  1. a simplex extends naturally to its real affine space which deformation retracts back to the simplex.
  2. a real affine extends to a complex affine space which deformation retracts back to the real affine space.
  3. now apply these two steps to any simplicial complex
  4. finite ones yield complex varieties defined over Q which deformation retract to any finite simplicial complex.
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