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?
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?”.