# Weight filtration for smooth analytic manifolds

In his ICM 2002 talk (Topology of singular algebraic varieties, available also on arXiv) B. Totaro says on p. 3 (of the arXiv version): "Using the method of Guillen and Navarro Aznar I was able to define the weight filtration for complex and real analytic spaces." However, no reference to this statement is given there. The definition is not explicitly given either, but from the preceding discussion a reasonable guess seems to be that one considers simply the Leray filtration of the open embedding into some compactification. The result will depend on the compactification, but if one compactification dominates another one, then the weight filtrations are the same (ibid, theorem 2.2).

I wasn't able to find a proof of the latter statement in the literature and I would like to ask if anyone knows a reference.

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Here's a link to the arxiv version arxiv.org/abs/math/0304296 –  j.c. Apr 14 '10 at 15:44

For a smooth complex algebraic variety X, the weight filtration is as you say the Leray filtration, up to reindexing, for a good compactification (the complement of should be a normal crossing divisor). For singular X, one needs to choose a diagram $X\leftarrow X_\bullet \to \bar X_\bullet$ such that the first arrow satisfies cohomological descent and the second arrow is a good compactification of a smooth simplicial (or cubic) scheme. See Deligne's Theorie de Hodge II, III, or Navarro Aznar et. al [Springer LNM 1335]. Although this approach hinges on certain Hodge theoretic facts that won't generalize (see comments), there is a refinement which I suspect will. Gillet-Soul\'e [Crelles 1996] showed that if one uses a slightly stronger descent property, then it determines a certain well defined complex $W(X)$ in the homotopy category of pure Chow motives. It is essentially the mapping cone $$Cone ([\bar X_\bullet]\to [\bar X_\bullet- X_\bullet])$$ where I write [...] for the complex of motives associated to a smooth (bi)simplicial scheme. $W(X)$ can be used to define an integral weight filtration on compactly supported cohomology. I think that Guillen-Navarro Aznar [IHES 2002] get a similar result.