Let $X$ be a projective variety (over $\mathbb{C}$). Is there a simplicial object $X_{\bullet}$ in the category of smooth projective varieties and a morphism of simplicial varieties $X_{\bullet} \rightarrow X$ inducing a weak homotopy equivalence (of spaces) $|X_{\bullet}| \rightarrow X$?

This (I believe) is claimed by Donu Arapura in a comment on the following question: Deligne's Mixed Hodge Theory

The statements I've seen about such resolutions (e.g. in Deligne's Hodge III) are all assertions about sheaf cohomology, and not homotopy. I'd be happy with just a reference, or an argument that this is a direct consequence of the statements I have alluded to.

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    $\begingroup$ I guess it's only implicit in Deligne. But look at, for example, Carlson, "Polyhedral resolutions of algebraic varieties" for a more explicit statement. I might say more when I have time. $\endgroup$ Jun 25, 2019 at 21:30

1 Answer 1


I'm converting my comment to an answer. Carlson, Polyhedral resolutions of algebraic varieties, shows to each projective variety $X$ there exists an augmented smooth (strict or semi) simplicial projective variety $X_\bullet\to X$ such that the induced map $|X_\bullet|\to X$ is a homotopy equivalence. You can formally add degeneracy maps to make it an actual simplicial object, if you require that.

Alternatively, you can look at Deligne's original construction in Hodge III or SGA4 (exp Vbis). This yields a hypercover $X_\bullet\to X$, i.e. $$X_{n+1}\to (cosk_n X_\bullet)_{n+1}$$ is proper and surjective for all $n$. This should imply the map $|X_\bullet|\to X$ is surjective with contractible fibres, as explained in section 8.5 of Artin-Mazur Etale homotopy. I leave to you to flesh out the details.


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