Why do we need localization by Leftschetz motive?

Question

Definition of the Grothendieck group and Leftschetz motive. The Grothendieck group of varieties is a free abelian group generated by classes of algebraic varieties with the following relation: $$ [X]=[X\smallsetminus Y]+[Y] $$ for $Y\subset X$ a closed subvariety. Let $\mathbb L$ denote the class of an affine line (the Leftschetz motive). For the base field $\mathbb C$ the Grothendieck group forms a ring ($[X\times Y]=[X]\times[Y]$).

Cancelation conjecture. This conjecture says the Lefschetz motive $\mathbb L$ is not a zero divisor in the Grothendieck ring, i.e., $\mathbb L^{-1}$ exists.

A counterexample. Paper [L. Borisov: Class of the affine line is a zero divisor in the Grothendieck ring, arXiv:1412.6194] provides two varieties $X$ and $Y$ (they are Calabi--Yau varieties with the same derived category of coherent sheaves, but they are not birationally equivalent) with the following relation in the Grothendieck ring: $$ ([X]-[Y])(\mathbb L^2-1)(\mathbb L-1)\mathbb L^7=0, $$ i.e., $\mathbb L$ is a zero divisor (of course, paper shows that all other factors are not zero).

There are papers that use the Cancelation conjecture, say [S. Galkin, E. Shinder: The Fano variety of lines and rationality problem for a cubic hypersurface, arXiv: 1405.5154]. My question is

(1) Is the Cancelation conjecture is important and why?

or, more precise,

(2) Which theorem and theories use the Cancelation conjecture? And which do not?

and also

(3) How does the results of Lev Borisov affect on these theories? Do we need the existence of $\mathbb L^{-1}$ or it is just formal requirement?

Maybe, my questions show my misunderstanding the difference between Motives and the Grothendieck group. I'll be glad if you correct me and help me understand the questions in both cases.

Answer 1

This is more of an opinion than a complete answer but it is too long to leave as a comment. Let $$H : K_0(\mathcal{M}_{\mathbb{C}}) \to K_0(MHM_{\mathbb{C}})$$ be the Hodge realization from the Grothendieck ring of Chow motives to the Grothendieck ring of mixed Hodge modules, then $H$ kills all $(\mathbb{L}-1)$-torsion.

However, a conjecture concerning a weight filtration on $\mathcal{M}_{\mathbb{C}}$ implies that $K_0(\mathcal{M}_{\mathbb{C}})$ does not have $(\mathbb{L}-1)$-torsion, and hence, the closure $\bar K$ of $K_0(Var_{\mathbb{C}})[\mathbb{L}^{-1}]$ in the completion of $K_0(Var_{\mathbb{C}})[\mathbb{L}^{-1}]$ along the dimensional filtration implies that the natural morphism $K_0(Var_{\mathbb{C}})[\mathbb{L}^{-1}]\to K_0(\mathcal{M}_{\mathbb{C}})$ factors through $\bar K$. Independently of whether this conjecture is true or not, the composition $K_0(Var_{\mathbb{C}})[\mathbb{L}^{-1}] \to K_0(MHM_{\mathbb{C}})$ does factor through $\bar K$.

Therefore, generically speaking, types of problems which concern stabilization of a sequence of elements in $K_0(Var_{\mathbb{C}})[\mathbb{L}^{-1}]$ giving rise to invariants in $K_0(MHM_{\mathbb{C}})$ and beyond will be unaffected. So, results concerning counting points or hodge numbers for example via ideas from motivic integration would remain valid.

So, if $\mathbb{L}$ is a zerodivisor, then the natural morphism $K_0(Var_{\mathbb{C}})[\mathbb{L}^{-1}]\to \bar K$ is not injective, yet this does not imply that the conjecture on a weight filtration of chow motives is wrong.

In the above, I am mostly following some remarks in the following two papers:

• J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Inventiones Mathematicae 135 (1999), 201-232.

• J. Denef and F. Loeser, Motivic Igusa zeta functions, Journal of Algebraic Geometry 7 (1998), 505-537.