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Deligne constructs Mixed Hodge Structures (MHS) on the cohomology, $H^{*}(X)$, of an algebraic variety $X$, in his papers Hodge II and Hodge III. Really the question below is rather vague, and is motivated by my desire to separate the construction into two parts, the linear algebraic and the geometric.

Much of the input is subtle linear/ homological algebra, ie deals with the target category of the construction (namely MHS). The geometric content, ie that dealing with the source category, seems to be really about approximating varieties by smooth projective such, and it's this I'd like to focus on.

If $U$ is a smooth variety the construction proceeds via an embedding $U\rightarrow X$ into a smooth projective $X$, so that the complement $D$ is normal crossings. One might imagine that this then determines what the MHS should be, at least if we demand Gysin sequences, however $D$ need not be smooth itself, and so we don't have a construction in this case. The construction for something of the form $D$ is gotten by taking a simplicial resolution of $D$, basically by taking the Cech nerve of the resolution of $D$ given by the disjoint union of its components.

I'd like then to say very roughly that the construction suggests that we should replace the category of varieties, $Var_{\mathbb{C}}$, with something like the category of simplicial objects in "closed inclusions of smooth projective varieties" ie level wise closed inclusions of simplicial smooth projective varieties. I'll call this category $C$ for now. I would love if there was a notion of weak equivalence in $C$, such that there was an inclusion of categories $Var_{\mathbb{C}}\rightarrow C_{loc} $, where the RHS denotes the localized category. I believe objects of $C$ should give rise to chain complexes of MHS and I want the weak equivalences to be taken to quasi-isomorphisms. Nb one must still check that the MHS corresponding to a variety is in the abelian categroy of MHS.

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    $\begingroup$ So your question is whether the construction can be reorganized this way? Yes, sort of. For example, if $X$ is a proper complex variety, then Deligne shows that there exists a simplicial object $X_\bullet$ in the category of smooth projective varieties such that $X$ is weakly homotopy equivalent to the geometric realization $|X_\bullet|$ (where everything has classical topology). Unfortunately, the last part of the story, to actually construct the MHS on $X$, will be more complicated than what you describe. $\endgroup$ – Donu Arapura Jun 17 at 0:58
  • $\begingroup$ @donu thanks for the comment! I guess that's all I can hope for. I was hoping for something that could treat the open and the proper case on equal footing, which is why I'd imagined taking simplicial objects in pairs of smooth proj. $\endgroup$ – EBz Jun 17 at 9:08
  • $\begingroup$ @donu, I suppose I'd like if some suitably localized category of simplicial varieties (maybe pairs thereof), was such that the inclusion of the full subcat of smooth proj such was an equivalence. I feel like this would make precise my (possibly wrong) intuition that "given a suitably nice target category there's a unique extension from smooth proj to all varieties". $\endgroup$ – EBz Jun 17 at 9:27
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    $\begingroup$ I suggest reading sections 2.2-2.6 in this paper of Beilinson: arXiv:1102.1294. $\endgroup$ – Denis-Charles Cisinski Jun 17 at 10:06
  • $\begingroup$ @Donu Arapura, is the statement about weak homotopy type easy to see from the results on cohomological descent? $\endgroup$ – rj7k8 Jun 17 at 23:50

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