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Let $X$ be a variety over $\mathbb{C}$, say separated. According to Deligne's results, there is a "mixed Hodge structure" on the total cohomology $H^\bullet(X(\mathbb{C}), \mathbb{Z})$. One component of this is a "weight filtration" on $H^\bullet(X(\mathbb{C}), \mathbb{Q})$. I haven't read Deligne's "Theorie de Hodge" and don't really understand all this, but I believe that in the case of a smooth projective variety, this reduces to usual Hodge theory and the weight filtration is the filtration by grading, and the extension to singular varieties comes by some sort of simplicial resolution by smooth objects.

Let $Y_0$ be a variety over a finite field $\kappa$. Given a mixed perverse sheaf $K_0$ on $Y_0$, there is a canonical (and functorial) weight filtration on $K_0$, such that the sucessive subquotients are pure complexes of increasing weight (in the sense of Weil II).

What do these to have to do with each other? In section 6 of BBD (asterisque 100), it seems that the authors are using the functoriality of the weight filtration over finite fields to deduce results about the weight filtration over $\mathbb{C}$. Namely, I'd be interested if, given a perverse sheaf $K$ (say, of geometric origin) on a smooth, proper scheme $X$ over $\mathbb{C}$ which can be "spread out" to perverse sheaves of "reduction of $X$ mod a prime*" there is some way in which the weight filtration on the cohomology of $K$ (actually, I'm not sure that this exists, it seems to in the constant case at least) can be viewed as a completion of the weight filtrations in finite characteristic.

Here is the specific result in BBD: Let $f: X \to Y$ be a separated morphism of schemes of finite type over $\mathbb{C}$. Suppose that the stalks of $R^n f_* \mathbb{Q}$ are $H^n(X_y, \mathbb{Q})$, and that these form a local system. Then the weight filtration on these stalks form a locally constant filtration of the local system $R^n f_* \mathbb{Q}$. This appears to be proved by reducing mod a prime, where one has a Frobenius and the perverse weight filtration makes sense.

(One reason to think these might be related is that if $X_0$ is a proper smooth scheme over $\mathbb{\kappa}$, then the cohomologies $H^i(X, \mathbb{Q}_l)$ have weight $i$ by the Weil conjectures, and this has some correspondence with how the weight filtration was defined for projective, smooth schemes over $\mathbb{C}$.)

*Which is done by reducing the field $\mathbb{C}$ of definition to some finitely generated ring over $\mathbb{Z}$, and then working from there.

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There is a theory of "Mixed Hodge Modules", due to Morihiko Saito, which is the precise Hodge theoretic analogue of the theory of mixed perverse shaves of BBD. It was, I think, motivated by BBD and gives alternative proofs of all the results of BBD for varieties over $\mathbb{C}$. For a survey, see: "Saito, Morihiko. Introduction to mixed Hodge modules. Actes du Colloque de Théorie de Hodge (Luminy, 1987). Astérisque No. 179-180 (1989), 10, 145–162". –  ulrich Aug 13 '11 at 15:27
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It follows from Saito's results that the cohomology of any perverse sheaf of geometric origin on a variety over $\mathbb{C}$ has a natural mixed Hodge structure, in particular, a weight filtration. It should be the same, after extension of scalars, as the one gotten by spreading out (as you described) but I don't know a reference where this is proved. –  ulrich Aug 13 '11 at 15:34
    
Thanks. I'll take a look at Saito's theory. –  Akhil Mathew Aug 14 '11 at 19:09
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Dear Akhil, One thing to bear in mind with Saito's theory is that it is very subtle, and my experience is that it is hard to figure out what is really going on. (The definition of what it means to be a mixed Hodge module is very indirect, relying on an analysis of iterated vanishing cycle constructions.) I am saying this just so that you don't have your expectations raised too high. I don't mean that you shouldn't look at it --- it is quite fundamental --- but just don't expect it to be easy to understand, or to provide immediate clarification. Best wishes, Matt –  Emerton Aug 14 '11 at 22:31
    
Dear Matt, thanks for the heads up! I'll keep that in mind when I do get around to learning the theory (and it seems I should first learn about vanishing cycles). –  Akhil Mathew Aug 14 '11 at 22:56
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2 Answers 2

up vote 8 down vote accepted

Dear Akhil,

This is a big topic, although one that has been discussed at various times here, e.g.

In what setting does one usually define mixed sheaves and weights for them?

The idea is that for constant coeffients smooth projective varieties should be cohomologically the simplest. And by approximating more general varieties by these, via simplicial techniques etc., we get a weight filtration on cohomology which measures the deviation from the simplest case.

How to make this precise? Well

  1. In positive characteristic, we can say that smooth projective varieties are one on which Frobenius acts with expected bounds eigenvalues. So the weight filtration is defined via eigenspaces of these.
  2. Over $\mathbb{C}$, smooth projective varietes carry classical Hodge decompositions. The weight filtration needs to be (nontrivially) inserted into this picture via mixed Hodge theory.

The compatibility of the weights comes either by construction* or via the (somewhat conjectural) story of mixed motives.

For perverse coefficients, the story is already much more complex. The "simplest" cases should be intersection cohomology complexes with coefficients in direct images of families of smooth projective varieties. The analogue of (1) is BBD, and of (2) is Saito's theory that Uhlirch mentions.


*(Added) Perhaps I can say what I mean "compatible by construction". I'll take two examples, which give a sense of what's going behind the scenes.

A) take $X$ to be the complement of two points $p,q$ in smooth projective curve $\bar X$. Then have an exact sequence $$ 0\to W_1= H_1(\bar X, \mathbb{Q}) \to W_2=H^1(X, \mathbb{Q})\to \mathbb{Q}(-1)\to 0$$ The last map can be thought of as sort of residue at $p$. The symbol $\mathbb{Q}(-1)$ means the one dim vector space shifted into weight $2$, so this sequence also displays the weight filtration, There is an entirely analogous sequence in the $\ell$-adic world which gives the weights there. So these are compatible (pretty much by design).

B) For the second example, let us use $\bar X$ as above but with coefficients in the intersection cohomology $L=j_\ast R^i f_\ast\mathbb{Q}$, where $f:Y\to X$ is smooth projective. Then $L$ carries variation of Hodge structure of weight $i$. By Zucker [Ann. Math 1979] $H^1(\bar X, L)$ has a pure Hodge structure of weight $1+i$. In the $\ell$-adic world, the analgous statement is Deligne's purity theorem [Weil II]. Note that Zucker's theorem was one of the key analytic inputs in Saito's work, analogous to the role of Weil II in BBD.

Some References: Matt is correct that Saito's work isn't easy to get into. Aside from some expositions by Saito, I might suggest looking at the last few chapters of Peters and Streenbrink's book on mixed Hodge theory, which gives a pretty good introduction. I'm also linking my own, not quite successful, attempt to go through some of this:

http://www.math.purdue.edu/~dvb/preprints/tifr.pdf

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Thanks for this answer! I'll have a look at Saito's theory of mixed Hodge modules (I'm still very far from understanding anything about motives). –  Akhil Mathew Aug 14 '11 at 3:10
    
Thanks to results by myself:) and David Hebert, those ingredients of the theory of mixed motives that are needed in order to generalize the long exact sequence above (to cohomology of arbitrary Voevodsky's motives, either over a field or over any excellent base scheme) are no longer conjectural. One defines a (Chow) weight structure for Voevodsky's motives; it yields certain weight spectral sequences that generalize classical ones. See arxiv.org/abs/0704.4003 arxiv.org/abs/1007.0219 arxiv.org/abs/1007.0219 –  Mikhail Bondarko Aug 15 '11 at 13:51
    
Mikhail, thanks for the information. –  Donu Arapura Aug 15 '11 at 14:21
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see Deligne's ICM 1970 address (Theorie de Hodge I) as well as his ICM 1974 address.

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