A new, very short paper with Michael Larsen, “Positive motivic measures are counting measures” is up on the arXiv today. I thought I’d say a bit here about where the problem came from, since we don’t do so in the paper.
In the project with Akshay that I talked about at the recent Columbia conference on rational curves on varieties, one thing you do is compute estimates for |M_n(F_q)|, where M_n is the moduli space of algebraic maps of degree n from P^1 to some fixed target variety X, and F_q is a finite field. These inequalities turn out to be very nicely uniform in q, which leads one naturally to ask; do the proofs actually give “motivic estimates” for the class [M_n] in the Grothendieck ring K_0(Var_K), for various non-finite fields K?
Well, what does it mean for one element r of a ring R to “estimate” another element s? It might mean that r-s is rather deep in some natural filtration on R. Those don’t seem to be the kind of estimates our methods provide; rather, they say something more like
(r-s)^2 <= B
where B is some fixed element of K_0(Var_K). But what does “<=” mean? Well, it means that B – (r-s)^2 is nonnegative. And what does “nonnegative” mean? That’s the question. What the proof really gives is that B – (r-s)^2 lies in a certain semiring N of “nonnegative motives” in K_0(Var_K). Let’s not be too precise about what N is; let’s just say that it includes [V] for every variety V, and it has the property that |n(F_q)| >=0 for all q, whenever n lies in N. In particular, that means that
(|r(F_q)| – |s(F_q)|)^2 <= B(F_q)
so that, on the level of counting points, s(F_q) really is a good estimate for r(F_q).
So one might ask: are there other interesting positive motivic measures — that is, homomorphisms
f: K_0(Var_K) -> reals
which take N to nonnegative reals? If so, f(s) would be a good estimate for f(r).
And the point of this note with Larsen is to say, with some regret, no — any motivic measure which assigns nonnegative values to the classes of varieties is in fact just counting points over some finite field. Which sort of kills in its crib the initial hope of some exciting world of “motivic inequalities.”
Of course, the reals are not the only ordered ring! As Bjorn Poonen pointed out to me, for a general field K you can find an ordered ring A and a measure
g: K_0(Var_K) -> A
which is positive in the sense that g sends every variety to an element of A greater than or equal to 0; these come from big ultraproducts of counting measures of different finite fields. Whether these measures are “interesting” I’m not sure.