As I tried to read up on motives in preparation for my last post, I thought of some questions that seemed natural to me, but weren’t addressed in the sources I was reading. So here they are, in the hope that experts will find them obvious. One set of questions concerns motives for nonproper varieties, the other concerns higher categorification.

## Motives for nonproper varieties

This is something that I haven’t found an official source for; I’m just putting together what I think I can deduce from what I’ve read. So be prepared for dumb errors.

Let be smooth, but not proper. In characteristic zero, we can always find a smooth normal crossings compactification of . Let be the divisors in the boundary.

I think that one defines a class , in the Grothendieck group of the category of motives, to be

By a result of Jannsen, the category of numerical equivalence motives is semi-simple, so we aren’t losing much data by passing to the Grothendieck group.

Using that any rational map is a composition of blowups and blowdowns, I think I have checked that this is not dependent on the choice of compactification. In this way, we assign an object in the Grothendieck group of the category of motives to a non-proper variety.

If has a stratification with strata , then .

As I understand it, this is the key to a lot of the uses of motives in the string-theory inspired literature. One notes that some quantity, such as point count (over a finite field), or Euler characteristic, or Hodge numbers (in characteristic zero), can be defined for a general motive. Then, given a stratification of , one can compute the quantity for by summing over the strata.

Some things that I don’t know: Is there a definition of the class that doesn’t use resolution of singularities? Is the notion known to be well defined in characteristic ?

There is an interesting object called the Chern-Schwartz-Macpherson class. For any smooth variety , this is a class in , which generalizes the notion of the Chern class for proper varieties, and adds nicely in decompositions. Does the motivic perspective explain why it exists?

## Higher categorification

This may just show that I spend too much time at the –Category cafe. But, to my mind, the following question seems irresistible. From the local systems perspective, cohomology is simply from the trivial local system to itself. If you are comfortable enough with the derived perspective (which I’m not, but I’ll pretend to be), you can just call these the derived endomorphisms of the trivial local system.

Is there some -category which is the “universal category of local systems”, and for which the motive of is the endomorphisms of the trivial local system on ? This would be especially interesting because there are several different ways of building the category of local systems, which tend to be related to different cohomology theories (etale, crystalline, etc.)

I have done no literature search, so this may be a very well studied topic. If it is, I’d be curious to know what key words to look under. If such a theory exists, it should be relevant to Emmanuel Kowalski’s excellent question “What is motivic Weil II?”.

Regarding your higher categorification, the category of motivic sheaves (on a given variety, or better, considered on all varieties, with an appropriate Grothendieck six operations formalism) may be what you are asking about.

(This is part of Beilinson’s conjectural picture of motives. It is to motives

themselves as variations of Hodge stucture are to Hodge structures.)

Regarding non-proper/non-smooth varieties, mixed motives is a relevant concept.

In my mind, the question I had asked was not so wide-ranging… As the previous comment says, it’s most likely that the motivic approach to Weil 2 should involve mixed motives, but what I have never seen is any kind of formal deduction of Deligne’s results from standard facts and conjectures about the latter. (Similar to the deduction of Weil 1 from the “standard conjectures” that you have sketched in the previous posts).

The two papers of Gillet and Soule, “Descent, Motives and K-theory” and “Motivic wight complexes for arithmetic K-theory” might be relevant for your question on motives of non-proper varieties. They prove much stronger results, dealing with Chow motives rather than Grothendieck motives, but Corollary 5.13 of the second implies a positive answer to the question of existence of a canonical element in the Grothendieck of motives (with rational coefficients since they use alterations) attached to any variety over any perfect field.

A few additional references :

For 1) (motives of non-proper varieties), the question of associating to a (non-necessarily proper or smooth) variety a class in the Grothendieck group of Chow motives was first asked by Serre. The answers by Gillet and Soulé were mentioned before, but there is another nice approach to the construction of weight complexes due to Bondarko. He associates such a class in K_0(Chow) to any motive in DM_k (with rational coefficients, k perfect), see . This hinges on the construction on a “weight structure” on DM_k, which is a triangulated category abstraction of the situation of pure motives inside mixed motives.

For 2) Higher categorification, to elaborate on what Emerton mentioned, there is a conjectural notion of motivic sheaf which would solve your problem : namely for every base scheme S, there should exist an abelian monoidal category MM(S) (actually one can speculate that there should be two variants : the standard – analoguous to constructible sheaves, e.g l-adic sheaves – and the perverse – analoguous to perverse sheaves , e.g mixed hodge modules or perverse l-adic sheaves) with the 6 operation fonctoriality at the derived category level, a notion of smooth objects or “motivic local systems” (equivalent to strong dualizability ?), realization functors to l-adic sheaves/mixed hodge modules, etc.

This is very much a dream, but the triangulated version of this is now pretty much developped (in part for rational coefficients only, and excepting the good notion of smooth object and the mixed hodge module realization). See for example the introduction to arXiv:0912.2110, which is I believe the state of the art on this issue.

Now, in this context, one should be careful that Hom_{DM_X}(1_X,1_X(p)[q]) is what people in this area call the motivic cohomology groups of X, and that their analogue in classical systems of coefficients are not geometric cohomology groups but rather “absolute” cohomology groups, like absolute hodge cohomology and continuous l-adic cohomology.

To wrap up question 1) and 2) together, a similar story of weight structure and weight complexes is now available for these triangulated categories of motivic sheaves over more general bases, see arXiv:1007.0219 and arXiv:1007.4543. In this context, pure motives are not as one might think the motives of proper smooth schemes over the base, but the motives of proper schemes with regular total space.