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By a generalized Euler characteristic $\chi$, I mean an isomorphism invariant $\chi(V)$ inside some abelian group $A$, defined for every varietiy $V$ over a field $k$, with the property that, for all varieties $X$ and open subvarieties $U$, we have $$\chi(X)=\chi(U)+\chi(X\setminus U).$$ (In other words $\chi$ is a group homomorphism $K_0(Var_k)\to A$ from the Grothendieck ring of varieties.)

There are many generalized Euler characteristics, for example the ordinary Euler characteristic (with compact support), the Hodge numbers (at least for smooth projective complex varieties), the number of rational points (over finite fields), etc.. However, all these examples are of motivic origin, i. e. they factorize over the map assigning to a smooth projective $X$ the class of its motive in the Grothendieck ring of Chow motives.

The only Euler characteristic I know of for which this is not the case is the map $K_0(Var_k)\to \Bbb Z[SB]$ where $SB$ denotes the monoid of stable birational equivalence classes, sending a smooth projective X to its equivalence class.

Are there any other known Euler characteristics factorizing neither through $K_0(Var_k)\to K_0(\mathcal Mot_k)$ nor $K_0(Var_k)\to \Bbb Z[SB]$?

(Here, of course, $\chi$ should be somewhat sensible, in the sense that the structure of target $A$ should not be to complicated (so as to exclude the universal Euler characteristic $id:K_0(Var_k)\to K_0(Var_k)$.)

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See section 7 of Bondal-Larsen-Lunt's paper "Grothendieck Ring of Pretriangulated categories, where they construct a non-trivial morphism $$\mu: K_0(\text{Var}_k)\to \mathcal{PT}$$ where the latter is, as you might guess from the title of their paper, a Grothendieck ring constructed from the $2$-category of pretriangulated categories. The idea is to send a variety to its derived category, though the paper is quite a bit more complicated than that.

This motivic measure factors through $$K_0(\text{Var}_k)/(\mathbb{L}-1);$$ hence it does not factor through $$\mathbb{Z}[SB]\simeq K_0(\text{Var}_k)/\mathbb{L}$$ (at least over an algebraically closed field of characteristic zero).

Orlov conjectures that derived equivalent varieties have isomorphic motives (which would imply this measure does factor throuh $K_0(\text{Mot}_k)$, I believe; this is very open, and I've heard some experts express skepticism about it--I'm agnostic, personally. At the very least, this motivic measure is not known to factor through $K_0(\text{Mot}_k)$, and some people doubt it does--if you could prove this, many people would be interested. This is the only measure I know of (off the top of my head) which is not known to factor through either $\mathbb{Z}[SB]$ or $K_0(\text{Mot}_k)$.

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    $\begingroup$ It's actually a conjecture of Orlov, see this paper. (The conjecture is for rational coefficients as it is clearly false integrally. I agree that it is not really expected to be true even rationally, though.) $\endgroup$ – AAK Dec 10 '15 at 10:06
  • $\begingroup$ It seems that the comparison of the non-commutative Chow motives (as defined by Tabuada?) with the "usual" ones yields that the aforementioned Orlov's conjecture is valid "modulo $\mathbb{L}-1$". $\endgroup$ – Mikhail Bondarko Dec 12 '15 at 13:15
  • $\begingroup$ @MikhailBondarko, that's right, it was also proved directly in the note of Orlov (the proof essentially amounts to the same thing as the comparison of noncommutative motives modulo Tate twists with usual motives, namely Grothendieck-Riemann-Roch). $\endgroup$ – AAK Dec 13 '15 at 15:12

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